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ELEMENTS   OP  PLANE   TRIGONOMETRY 


A  SERIES  OF  MATHEMATICAL  TEXTS 

EDITED    BY 

EARLE  RAYMOND  HEDRICK 


THE   CALCULUS 

By   Ellery    Williams    Davis    and    William    Charles 
Brenke. 
ANALYTIC   GEOMETRY   AND   ALGEBRA 

By  Alexander  Ziwet  and  Louis  Allen   Hopkins. 
ELEMENTS   OF   ANALYTIC   GEOMETRY 

By  Alexander  Ziwet  and  Louis  Allen  Hopkins. 

PLANE     AND     SPHERICAL     TRIGONOMETRY     WITH 
COMPLETE   TABLES 

By  Alfred  Monroe  Kenyon  and  Louis  Ingold. 
PLANE     AND     SPHERICAL     TRIGONOMETRY     WITH 
BRIEF  TABLES 

By  Alfred  Monroe  Kenyon  and  Louis  Ingold. 
ELEMENTARY   MATHEMATICAL   ANALYSIS 

By  John  Wesley  Young  and  Frank  Millett  Morgan. 
COLLEGE   ALGEBRA 

By  Ernest  Brown  Skinner. 
ELEMENTS  OF   PLANE   TRIGONOMETRY    WITH   COM- 
PLETE  TABLES 

By  Alfred  Monroe  Kenyon  and  Louis  Ingold. 
ELEMENTS  OF  PLANE  TRIGONOMETRY  WITH  BRIEF 
TABLES 

By  Alfred  Monroe  Kenyon  and  Louis  Ingold. 
THE   MACMILLAN  TABLES 

Prepared  under  the  direction  of  Earle  Raymond  Hedrick. 
PLANE   GEOMETRY 

By  Walter  Burton  Ford  and  Charles  Ammerman. 
^LANE   AND   SOLID   GEOMETRY 

By  Walter  Burton  Ford  and  Charles  Ammerman. 
SOLID   GEOMETRY 

By  Walter  Burton  Ford  and  Charles  Ammerman. 
CONSTRUCTIVE   GEOMETRY 

Prepared  under  the  direction  of  Earle  Raymond  Hedrick. 
JUNIOR   HIGH   SCHOOL   MATHEMATICS 

By  W.  L.  VosBURGH  and  F.  W.  Gentleman. 


This  book  is  issued  in  a  form  identical  with  that  of  the  books  announced  above 


ELEMENTS  OF 
PLANE   TRIGONOMETRY 


BY 
ALFRED   MONROE   KENYON 

PROFESSOR    OF    MATHEMATICS,    PURDUE    UNIVERSITY 
AND 

LOUIS   INGOLD 

ASSISTANT    PROFESSOR    OF    MATHEMATICS 
THE    UNIVERSITY    OF    MISSOURI 


THE   MACMILLAN   COMPANY 
1921 

All  rights  reserved 


GOPYBIGHT,   1919, 

By  the  MACMILLAN  COMPANY. 


Set  up  and  electrotyped.    Published  April,  1919. 


ASTRONOvy  0!!Pt>. 


Norinooli  i^resg 

J.  8.  Cashing  Co.  —  Berwick  &  Smith  Co. 

Norwood,  Mass.,  U.S.A. 


PREFACE 


This  book  carries  out  the  chief  motives  which  guided  the 
authors  in  their  larger  work  on  Plane  and  Spherical  Trigonom- 
etry. On  the  other  hand  it  has  been  entirely  rewritten,  and 
has  been  made  still  more  elementary  in  character.  The  new 
text  forms  a  treatment  of  Plane  Trigonometry  which  is  quite 
brief,  but  which  nevertheless  deals  with  the  most  essential 
topics  in  more  than  the  usual  detail. 

This  has  been  accomplished  by  omitting  or  curtailing  certain 
topics  that  are  seldom  used  by  the  student  except  in  some 
special  line  of  work.  Thus  all  of  Spherical' Trigonometry  and 
much  of  the  detailed  discussion  of  Trigonometric  Identities 
and  Equations  is  omitted.  Such  traditional  topics  as  De 
Moivre's  Theorem  and  infinite  series  were  omitted  from  the 
author's  larger  work  because  they  have  few  applications  with- 
in the  student's  present  grasp.  These  are  of  course  omitted 
from  the  present  book  also. 

Thus  this  treatment  contains  a  minimum  of  purely  theoreti- 
cal matter.  Its  entire  organization  is  intended  to  give  a  clear 
view  of  the  immediate  usefulness  of  trigonometry. 

The  solution  of  Triangles  remains  the  principal  motive.  As 
such,  this  problem  is  attacked  immediately  and  it  is  pushed 
to  a  definite  conclusion  early  in  the  course. 

More  complete  outlines  than  usual  have  been  given  for  the 
solution  of  oblique  triangles  by  means  of  right  triangles.  This 
method  of  solution  was  emphasized  recently  in  the  Syllabus  of 
the  War  Department  for  instruction  in  the  S.  A.  T.  C.  A  very 
brief  course  could  well  close  with  this  method  of  solving  tri- 
angles. 

Other  practical  problems  are  introduced  to  furnish  a  motive 
for  the  treatment  of  the  general  angle,  the  addition  theorems, 
radian  measure,  etc.     Among  other  applications,  the  composi- 


vi  PREFACE 

tion  and  resolution  of  forces,  projections,  and  angular  speed 
are  introduced  prominently. 

The  tables  are  very  complete  and  usable.  Attention  is 
called  particularly  to  the  table  of  squares,  square  roots,  cubes, 
etc. ;  by  its  use  the  Pythagorean  theorem  and  the  cosine  law 
become  practicable  for  actual  computation.  The  use  of  the 
slide  rule  and  of  four-place  tables  is  encouraged  for  problems 
that  do  not  demand  extreme  accuracy.  One  edition  of  the  book 
contains  only  the  four-place  tables.  Many  who  use  that  edi- 
tion find  it  advisable  to  have  students  purchase  also  the  five- 
place  tables  which  are  published  separately  bound  under  the 
title  The  Macmillan  Tables. 

The  authors  have  borne  in  mind  constantly  the  needs  of  the 
beginner  in  trigonometry  and  have  adapted  the  book  to  use  in 
secondary  schools  as  well  as  in  colleges.  Illustrative  mate- 
rial abounds,  and  the  explanations  have  been  carefully 
worked  out  in  great  detail.  The  sample  forms  for  the  solu- 
tion of  triangles  is  a  striking  instance  of  this  tendency. 

A.  M.  Kenyon. 
Louis  Ingold. 


CONTENTS 

PART  I.     ACUTE   ANGLES  AND  RIGHT  TRIANGLES 


Chapter  I.     Introduction 

§  1.  Subject  Matter 

§  2.  Measurement  

§  3.  Relations  to  Other  Subjects.     Applications 

§  4.  Graphical  Solution  of  Triangles 

§  5.  Preliminary  Estimate.     Check 

§  6.  Measurements  in  the  Field 

§  7.  Angles  of  Elevation  and  Depression 

§  8.  Squared  Paper 

§  9.  Rectangular  Coordinates 


Chapter 
§10. 
§11. 
§12. 
§13. 
§14. 
§15. 
§16. 
§17. 
§18. 
§19. 

Chapter 

§20. 
§21. 
§22. 
§23. 
§24. 
§25. 
§26. 
§27. 

Chapter 

§28. 
§29. 


11.     Definitions  —  Solution  op  Right  Triangles 
Tables       ...... 

Definitions  of  the  Ratios 

Right  Triangles         .... 

Elementary  Relations 
Construction  of  Small  Tables  . 
Functions  of  Complementary  Angles 

Apphcations 

Directions  for  Solving  Triangles 
The  Question  of  Greater  Accuracy 
The  Use  of  the  Large  Tables    . 

in.     Trigonometric  Relations 
Introduction 


1 
1 
2 
2 
4 
6 
8 
9 
10 

13 
13 
15 
15 
17 
18 
19 
22 
23 
24 

27 

Pythagorean  Relations 27 

Functions  of  O"*  and  90° 29 

Functions  of  30°,  45°,  60° 29 

Trigonometric  Equations 30 

Inverse  Functions 31 

Projections 34 


IV.     Logarithmic  Solutions  of  Right  Triangles 

The  Use  of  Logarithms 

Products  with  Negative  Factors       .... 

vii      ; 


37 

38 


viii  CONTENTS 


PAOK 


Chapter  V.     Solution  of  Oblique  Angles  by  Means  of  Right 
Triangles 
§  30.     Decomposition  of  Oblique  Triangles  into  Right  Triangles      42 
§  31.     Case  I:  Given  Two  Angles  and  a  Side      ....      43 
§  32.     Case  II:  Given  Two  Sides  and  the  Included  Angle  .       44 

§  33.     Case  III:  Given  the  Three  Sides 45 

§  34.     Case  IV:  Given  Two  Sides  and  the  Angle  Opposite  One 

of  Them 45 


PART  n.     OBTUSE   ANGLES   AND   OBLIQUE   TRIANGLES 

Chapter  VI.     Fundamental  Definitions  and  Formulas 

§  35.     Obtuse  Angles 49 

§  36.  Reduction  from  Obtuse  to  Acute  Angles          ...  50 

§  37.     Geometric  Relations 51 

§  38.     The  Law  of  Cosines  . 51 

§  39.    The  Law  of  Sines 53 

§  40.     Diameter  of  Circumscribed  Circle 54 

§  41.     The  Law  of  Tangents 55 

§  42.  Tangents  of  the  Half-angles      .        ^         ....  57 

§  43.    Radius  of  the  Inscribed  Circle 58 

Chapter  VII.     Systematic  Solution  of  Oblique  Triangles 

§  44.     Analysis  of  Data 60 

60 
62 
63 
65 
66 
68 


§  45.  Case  I:  Given  Two  Angles  and  a  Side 

§  46.  Case  II:  Given  Two  Sides  and  the  Included  Angle 

§  47.  Logarithmic  Solution  of  Case  II       .         .        . 

§  48.  Case  III :  Given  the  Three  Sides     . 

§  49.  Logarithmic  Solution  of  Case  III     . 

§  50.  Case  IV:  The  Ambiguous  Case 

Chapter  VEIL     Areas  —  Applications  —  Problems 

§  51.     Areas  of  Triangles  72 

§  52.  Area  from  two  Sides  and  the  Included  Angle          .        .  72 

§  63.     Area  from  Three  Sides 72 

§  54.     Illustrative  Examples 73 

§  55.  Composition  and  Resolution  of  Forces  and  Velocities      .  75 

§  56.    Illustrative  Examples 76 

PART  III.     THE    GENERAL  ANGLE 

Chapter  IX.     Directed  Angles  —  Radian  Measure 

§  57.     Directed  Lines  and  Segments 82 

§  58.     Rotation.     Directed  Angles 82 


CONTENTS  ix 

PAGE 

§  69.     Placing  Angles  on  Rectangular  Axes       .         ...  83 

§  60.     Measurement  of  Angles 85 

§  61.     Radian  Measure  of  Angles        ......  85 

§  62.     Use  of  Radian  Measure 85 

§  63.     Angular  Speed 86 

§  64.     Notation 86 

Chapter  X.     Functions  of  Any  Angle 

§  66.     Resolution  of  Forces.     Projections           ....  89 
§  66,     General  Definitions.      Trigonometric  Functions  of  Any 

Angle 89 

§  67.     Algebraic  Signs  of  the  Trigonometric  Functions      .         .  91 
§  68.     Reading   of    the  Tables.       Functions  of  -  6>,    90°  +  <?, 

180^  ±  e,  270°  ±e 93 

§  69.     Solution  of  Trigonometric  Equations        .         .         .         .95 
§  70.     Illustrative  Examples  on  Composition  and  Resolution  of 

Forces 96 

Chapter  XI.     The  Addition  Formulas 

§  71.     The  Addition  Formulas 98 

§  72.    The  Subtraction  Formulas 99 

§73.     Reduction  of  J.  cos  a  db  ^  sin  oc 99 

§  74.     Double  Angles 101 

§  76.    Tangent  of  a  Sum  or  Difference 101 

§  76.     Applications 102 

§  77.     Functions  of  Half  Angles 104 

§  78.     Factor  Formulas 106 

Chapter  XII.     Graphs  of  Trigonometric  Functions 

§  79.     Scales  and  Units 109 

§  80.     Plotting  Points .109 

§  81.     Graph  of  sin  aj 109 

§  82.     Mechanical  Construction  of  the  Graph     .         .         .         .111 

§  83.     Inverse  Functions 114 

§  84.     Graphical  Representation  of  the  Inverse  Functions          .  114 

LOGARITHMIC    AND   TRIGONOMETRIC   TABLES 

[See  Contents,  page  xviii.] 


ELEMENTS    OF 
PLANE    TRIGONOMETRY 

PART  I.     ACUTE  ANGLES  AND   RIGHT 
TRIANGLES 

CHAPTER   I 
INTRODUCTION 

1.  Subject  Matter.  The  word  Trigonometry  comes  from 
two  Greek  words  meaning  measurement  of,  or  by  means  of, 
triangles.  The  original  purpose  of  this  study  was  the  meas- 
urement of  angles  and  distances  by  indirect  methods  in  cases 
in  which  direct  measurements  are  inconvenient  or  impossible. 
Among  such  cases  we  may  mention  the  determination  of  the 
heights  and  horizontal  widths  of  hills,  the  distance  across 
a  valley  or  river,  or  the  lengths  of  the  boundaries  of  fields 
on  rough  or  impassable  ground.  Trigonometry  treats  also 
the  relations  among  the  sides  and  angles  of  triangles,  and  the 
measurement  of  the  sides,  angles,  and  areas  of  triangles  and 
of  other  polygons  which  can  be  separated  into  triangles. 

2.  Measurement.  To  measure  any  quantity  is  to  deter- 
mine how  many  times  it  contains  some  convenient  unit 
quantity  of  the  same  kind.  The  expression  of  every  measured 
quantity  consists  of  these  two  components:  the  numerical 
measure  and  the  name  of  the  unit  employed ;  as,  2  inches, 
20  cubic  centimeters,  3  pounds  and  10  ounces,  7  hours  and 
26  minutes,  51.72  acres,  36  degrees,  7.4  feet  per  second, 
35.8  ohms,  2.3  amperes,  110  volts,  etc. 

B  1 


PLANE  TRIGONOMETRY 


[I,§2 


Sometimes  we  can  make  direct  comparison  of  a  quantity  with 
the  unit  of  measure,  as  when  we  determine  the  length  of  a 
segment  by  applying  a  yardstick  or  a  steel  tape  to  it.  On  the 
other  hand  we  are  often  obliged  to  use  indirect  methods,  i,e. 
to  compute  the  numerical  measure  of  a  quantity  by  means  of 
its  relations  to  other  quantities  more  easily  measured.  Thus, 
we  find  the  numerical  measure  of  the  area  of  a  triangle  not  by 
direct  measurement,  but  rather  by  taking  one-half  the  product 
of  the  numerical  measures  of  its  base  and  its  altitude. 


3.  Relations  to  Other  Subjects.  Applications.  It  is  evi- 
dent that  trigonometry  is  closely  related  to  plane  geometry 
on  account  of  its  use  of  lines,  angles,  triangles  and  other 
polygons.  On  the  other  hand,  since  the  measures  of  the  sides, 
angles,  and  areas  of  triangles,  and  the  ratios  of  the  sides,  are 
numbers,  trigonometry  is  also  related  to  arithmetic  and  ele- 
mentary algebra. 

The  applications  of  trigonometry  are  very  extensive.  Some 
of  them  will  be  given  in  this  book.  Many  others  are  to  be 
found  in  surveying,  navigation,  astronomy,  architecture,  design, 
geometry,  mechanics,  and  other  branches  of  mathematics  and 
physics,  and  in  military  and  civil  engineering. 

4.  Graphical  Solution  of  Triangles.  For  constructing  tri- 
angles and  measuring  their  parts,  the  student  should  have  a 


Fig.  1. 


I,  §4] 


INTRODUCTION 


scale  for  measuring  lengths,  a,  protractor  for  measuring  angles,  and 
a  compass  for  drawing  circles,  laying  off  arcs  and  equal  segments. 

Two  triangles,  or  other  geometric  figures,  are  said  to  be 
congruent  when  they  can  be  superimposed  so  as  to  coincide  in 
all  their  parts. 

Two  figures  are  similar  when  their  corresponding  angles  are 
equal  and  their  corresponding  sides  are  proportional.  Two 
triangles  are  similar  if  they  are  mutually  equiangular,  but  this  is 
not  necessarily  true  of  polygons  of  more  than  three  sides. 


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Fig.  2. 

To  draw  a  figure  to  scale  is  to  make  a  drawing  which  shall 
be  similar  to  it  but  smaller  (or  larger),  as,  for  example,  a  map 
of  a  farm  or  a  field,  or  the  floor 
plan  of  a  building. 

The  advantage  of  a  scale 
drawing  is  that  the  angles  are 
the  same  as  those  of  the  figure 
represented,  and  by  the  scale 
relation  marked  on  the  drawing, 
any  dimension  of  the  original 
figure  can  be  read  off  on  a 
scale  applied  to  the  correspond- 
ing dimension  of  the  draw- 
ing. 

A  builder  uses  the  architect's  plans  for  this  purpose  in  con- 
structing a  building. 


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Fig.  3. 


4  PLANE  TRIGONOMETRY  [I,  §5 

We  know  from  geometry  that  the  other  three  parts  of  any 
actual  *  triangle  are  determined  if  any  one  of  the  following  com- 
binations is  known : 

(1)  tivo  sides  and  the  included  angle; 

(2)  tivo  angles  and  any  specified  side; 

(3)  the  three  sides; 

(4)  two  sides  and  the  a^igle  opposite  one  of  them, 

but  in  the  last  case  there  may  be  two  solutions  when  the  given 
angle  is  acute. 

When  a  sufficient  number  of  parts  of  an  actual  *  triangle 
are  known,  the  others  can  be  found  by  drawing  the  triangle  to 
scale  and  measuring  the  sides  with  the  scale  and  the  angles 
with  the  protractor. 

The  process  of  finding  the  unknown  parts  of  a  triangle  from 
any  such  set  of  given  parts  is  called  solving  the  triangle. 

Example  1.  In  order  to  measure  the  width  of  a  river,  for  example,  it 
is  sufficient  to  measure  the  distance  AB  between  two  points  on  the  bank 
and  the  angles  BAP  and  ABP  made  by  AB  with 
the  lines  joining  A  and  J5,  respectively,  to  any 
point  on  the  other  bank.  All  of  these  measure- 
ments can  be  made  from  one  bank  of  the  river. 
Knowing  AB  and  the  angles  ABP  and  BAP  the 
triangle  PAB  can  be  drawn  to  scale  ;  then  the 
^^^'  *•  perpendicular  PB  from  P  to  AB  can  be  drawn 

and  measured,  whence  the  width  PR  of  the  stream  can  be  determined  by- 
actual  measurement  in  the  figure.  If  J.B  =  98  yards,  AA  =  38°,  and 
Z.B=  65°,  PB  will  be  found  to  be  about  56  yards. 

5.  Preliminary  Estimate.  Check.  In  every  exercise,  the 
student  should  make  a  preliminary  estimate  of  the  unknown 
parts  and  he  should  keep  this  crude  solution  in  mind  to  guide 
him  in  his  work. 

After  the  unknown  parts  have  been  found,  the  student 
should  use  all  means  at  his  command  to  check  each  answer, 


♦The  data  can  be  given  so  that  it  will  be  impossible  to  construct  any 
triangle  satisfying  the  conditions.  If  such  data  are  given,  the  impossibility 
will  appear  when  the  attempt  to  construct  the  triangle  is  made. 


I,  §  5]  INTRODUCTION  5 

since  even  experienced  persons  are  liable  to  error  in  reading 
scales  and  in  making  computations. 

In  triangles  drawn  to  scale  observe  the  following  checks  : 

(1)  the  sum  of  the  angles  of  any  triangle  should  he  180° ; 

(2)  the  sum  of  any  two  sides  shoidd  he  greater  than  the 
third  side  ; 

(3)  the  greater  of  two  sides  should  he  opposite  the  greater  of  the 
angles  opposite  these  sides ; 

(4)  if  two  sides  are  unequal  their  numerical  measures  shoidd  he 
unequal  in  the  same  sense; 

(5)  the  numerical  measures  of  angles  should  correspond  to  their 
magnitudes;  angles  of  30°,  45°,  60°,  90°,  etc.,  are  easy  to  judge 
by  the  eye. 

These  checks  should  reveal  any  gross  error ;  but  the  student 
should  not  expect  this  method  of  solution  (or  any  other  method 
of  computation  or  measurement)  to  give  precise  answers  in  the 
sense  of  having  no  error  whatever.  The  purpose  should  be.  to 
obtain  reasonably  accurate  results  and  to  detect  errors  that  are 
unreasonably  large, 

EXERCISES  I.  — GRAPHICAL  SOLUTION  OF  TRIANGLES 

Solve  the  following  triangles  by  construction  and  measurement. 

1.  Two  angles  are  4t1°  and  53"^  and  the  included  side  is  5.7 

Ans.  80°,  4.2,  4.6 

2.  Two  angles  are  43°  and  53°  and  the  side  opposite  the  latter  is  6. 7 

Ans,  84°,  5.7,  8.3 

3.  Two  sides  are  4.3  and  5.3  and  the  included  angle  is  57°. 

Ans,  61°,  72°,  4.7 

4.  The  three  sides  are  4.3,  5.3,  and  6.3  Ans,  42°,  56°,  81°. 

5.  Two  angles  are  40°  and  65°  and  the  side  opposite  the  latter  is  50. 

Ans.  75°,  35.5,  53.3 

6.  Two  angles  are  30°  and  105°  and  the  included  side  is  7  feet  8  inches. 

Ans.  45°,  5  ft.,  9.7  ft. 

7.  Two  sides  are  16.9  and  40.9  and  the  altitude  upon  the  third  side  is 
12.     Find  the  perimeter  and  the  area.  Ans.  108.8,  306. 

8.  Two  angles  are  30°  and  100°  and  the  shortest  side  is  8.  Find  the 
longest  side,  the  altitude  upon  it,  and  the  area.         Ans.  15.8,  6.1,  48.2 


6  PLANE   TRIGONOMETRY  [I,  §5 

9.   The  sides  are  in  the  ratio  3:4:5.     Find  the  smallest  and  the 
largest  angle.  Ans.  37°,  90°. 

10.  The  angles  are  in  the  ratio  3:4:5  and  the  shortest  side  is  30. 
Find  the  other  sides.  Ans.  37,  41. 

11.  The  sides  are  5,  7,  and  8.     Find  the  angles.     Ans.  38°,  60°,  82°. 

12.  The  sides  are  3,  5,  and  7,     Find  the  largest  angle.       Ans.  120°. 

13.  Two  sides  are  8  and  10  and  the  included  angle  is  47°.  Find  the 
perimeter,  the  area,  and  the  radius  of  the  inscribed  circle. 

Ans.  25.4,  29.25,  2.3 

14.  From  which  of  the  following  sets  of  given  parts  is  it  possible  to 
construct  a  triangle  ?    Do  any  of  the  sets  det^ermine  more  than  one  ? 

(a)  Two  angles  are  41°  and  59°,  the  side  opposite  the  latter  is  5.1 
(6)  Two  sides  are  1.3  and  5.6,  the  angle  opposite  the  first  is  66°. 

(c)  Two  angles  are  30°  and  41°,  the  included  side  is  7. 

(d)  Two  sides  are  7  and  1.1,  the  included  angle  is  17°. 

(e)  The  three  sides  are  1.1,  2.3,  3.5 

(/)  Two  sides  are  6  and  7,  the  angle  opposite  the  first  is  51°. 

Ans.  (6)  and  (e),  impossible  ;  (/),  two. 

15.  Two  sides  are  5  and  7  and  the  angle  opposite  the  latter  is  60°. 
Find  the  perimeter  and  the  area.  Ans.  20;  17.3 

6.  Measurements  in  the  Field.  In  surveying  land,  rivers, 
lakes,  and  harbors ;  laying  out  roads,  ditches,  the  foundations 
of  bridges,  buildings,  and  other  structures  ;  and  in  many  other 
projects  of  civil  and  military  engineering,  distances  in  the 
field  are  measured  with  the  chain,  or  the  steel  tape.  In  cases 
where  extreme  accuracy  is  required,  a  long  metal  or  wooden 
scale  is  used,  and  is  carefully  protected  against,  and  corrected 
for,  changes  in  temperature. 

Angles  in  the  horizontal  plane  are  drawn  in  position  on  the 
plane  table  by  means  of  a  pair  of  sights  on  a  heavy  metal 
straightedge ;  or,  more  often  both  horizontal  and  vertical 
angles  are  sighted  with  the  telescope  of  the  engineer's  transit 
and  their  measures  are  read  off  from  the  graduated  circles  of 
the  instrument. 

In  determining  distances  and  directions  in  an  extended  survey,  greater 
accuracy  can  be  attained  by  measuring  the  angles  of  certain  triangles  and 
computing  the  lengths  of  the  sides,  than  by  measuring  these  sides  directly. 


I,  §6] 


INTRODUCTION 


Fig   5. 


A  base  line  AB  is  first  established  and  measured  with  great  precision. 
Then  some  point  C,  visible  from  both  A  and  B^  is  selected  and  the  angles 
CAB  and  ^50  are  measured  ;  another 
point  D  is  next  selected  and  the  angles 
CBD  and  BCD  are  measured.  Thus, 
a  chain  of  triangles  can  be  extended 
over  a  wide  range  of  territory  and  on  ^ 
completing  the  computations  the  length 
and  direction  of  every  line  in  the  sys- 
tem will  be  known.  This  process, 
called  triangulation^  is  used  by  the 
U.  S.  Coast  and  Geodetic  Survey. 
Much  work  has  been  done  near  the  coasts  and  a  triangulation  system  has 
been  extended  from  the  Atlantic  to  the  Pacific. 


Fig.  6. 


8  PLANE  TRIGONOMETRY  [I,  §7 

7.  Angles  of  Elevation  and  Depression.  An  observer  at  O 
measures  tlie  angle  of  elevation  of  an  object  A,  higher  than 
himself,  by  sighting  a  horizontal  line  OH  by  means  of  the 
level  on  the  telescope  of  the  transit  and  then  elevating  the 
end  of  the  telescope  until  he  sights  A,  The  angle  HOA 
through  which  the  telescope  has  been  turned  in  the  vertical 
plane,  and  which  is  read  off  from  the  vertical  graduated  circle 
of  the  transit,  is  the  angle  of  elevation  of  the  object  A  above 


Horizontal  Line 

Fig.  7. 

the  observer  at  O.  Similarly  he  measures  the  angle  of  de- 
pression of  an  object  B,  lower  than  himself,  by  first  sighting 
the  horizontal  line  OH  and  depressing  the  end  of  the  telescope 
through  the  angle  HOB  until  he  sights  B, 

EXERCISES  II.— GRAPHICAL  SOLUTION  OF  TRIANGLES 

Solve  the  following  exercises  by  construction  and  measurement. 

1.  Two  sides  of  a  triangular  field  are  70.6  rods  and  140.5  rods  and  the 
angle  opposite  the  latter  is  40°.     Find  the  length  of  the  fence  around  it. 

Ans.  353.9  or  529.6 

2.  At  a  point  in  the  street  midway  between  two  buildings  their  angles 
of  elevation  are  30°  and  60°  respectively.     Find  the  ratio  of  their  heights. 

Am,  1  : 3. 

3.  The  hands  of  a  clock  are  4  and  6  inches  long  respectively.  Find 
the  distance  between  their  tips  at  6  :  10  o'clock.  Ans.  6.3 

4.  In  the  triangle  ABG^  angle  A  =  64°,  B  =  72°,  and  the  included  side 
is  14.  Find  (a)  the  angle  at  the  center  of  the  circumscribed  circle  sub- 
tended by  the  side  AB  ;  (6)  the  angle  at  the  center  of  the  inscribed  circle 
subtended  by  BC;  (c)  the  length  of  the  altitude  from  G  upon  AB, 

Ans.  88°,  122°,  17.2 
6.   The  diagonals  of  a  parallelogram  are  10  and  12  and  they  cross  at 
an  angle  of  45°.    Find  the  sides.  Ans.  4.3,  10.1 


I,  §8] 


INTRODUCTION 


6.  The  steps  of  a  stairway  have  a  tread  of  10  in.  and  a  rise  of  7  in.; 
at  what  angle  is  the  stairway  inclined  to  the  floor  ?  Ans.  35'^. 

7.  Two  sides  of  a  triangle  are  each  6  and  the  included  angle  is  120°. 
Find  the  perimeter  and  the  area.  Ans.  22.4,  15.6 

8.  Find    the    distance    PQ    across    the    pond 
(Fig.  8)  from  the  following  measurements,  AP  z 
900  tt.,AQ  =  780  ft.,  PAQ  =  48°.         Ans.  692. 

9.  To  determine  the  width  AB  of  a  hill,  a  point 
G  is  taken  from  which  the  points  A  and  B  on  op- 
posite sides  of  the  hill  are  visible.  If.  AC  =  200  ft., 
BC  =  22S  ft.,  and  angle  J.C5  =  62°,  find  the  width 
AB. 

10.  The  angles  of  a  triangle  are  in  the  ratio  1:2:3,  and  the  altitude 
upon  the  longest  side  is  37.5.     Find  the  perimeter  and  the  area. 

Ans.  204.9,  1623.75. 

11.  Find  the  angles  and  sides  of  a  regular  five-pointed  star  inscribed 
in  a  circle  of  radius  10.  Ans.  36°,  19. 

8.   Squared  Paper.     It  is  often  an  advantage  to  draw  the 
figure  on  paper  ruled  into  squares,  called  squared  paper,  or 


P 

/ 

V 

/ 

> 

\ 

/ 

/ 

\ 

A 

/ 

\ 

1 

/ 

\ 

/ 

/" 

\ 

y 

/ 

c°, 

A 

/I 

^\ 

38 

R 

i 

\ 

B 

98 

1 

s^ 

ace 

2J^ 

\jl 

s. 

Fig.  9. 


cross-section  paper.  The  location  of  points  is  particularly- 
easy  on  such  paper,  so  that  a  map,  for  example,  is  readily 
made  by  using  it.  By  suitably  placing  the  figure,  required 
lengths  can  frequently  be  read  off  at  once. 

Thus,  if  the  triangle  for  the  graphical  solution  of  Ex.  1,  §  4,  be  con- 
structed on  cross-section  paper,  the  required  distance,  PB.,  Fig.  9,  can  be 
seen  at  once  to  be  about  56  yards. 


10 


PLANE  TRIGONOMETRY 


[I,  §9 


9.  Rectangular  Coordinates.  If  any  two  perpendicular 
rulings  OT  and  OX  of  the  squared  paper  (see  Fig.  10)  are 
selected,  the  position  of  any  point  P  in  the  plane  is  determined 
by  means  of  the  distances  from  these  two  lines  to  the  point  P, 
The  paper  can  be  so  placed  that  these  distances  are  vertical 
and  horizontal,  respectively;  we  shall  usually  suppose  the 
paper  in  this  position. 


^ 

"~ 

"~ 

"~" 

~* 

y 

^Y 

■" 

^ 

~" 

^ 

)^ 

~" 

■"" 

-1- 

2, 

1-.^ 

)- 

.B 

(- 

1, 

l.;2)^ 

SECiOND 

QUA 

DRANT 

f-;iRp 

Quadra:nt 

^ 

3 

- 

■X- 

^ 

0 

7 

1 

UnitJ 

TH 

RD  qu'adr; 

NT 

FOU 

RT 

H 

QUA.DF 

A[^T 

P 

1 

(1.4,- 

-.8)^ 

,c 

\' 

-1 

— 

1) 

Fig.  10. 


Thus,  in  Fig.  10,  the  horizontal  distance  from  OF  to  the  point  ^  is  1.2 
units.  To  avoid  confusion  between  points  at  the  same  distance  above  (or 
below)  OX  but  on  opposite  sides  of  OY,  it  is  customary  to  call  distances 
measured  to  the  right  of  OF  positive,  distances  to  the  left  of  OF  negative  ; 
thus,  B  is  said  to  be  —1  unit  from  OY.  Similarly,  distances  measured 
downwards  from  OX  are  called  negative  ;  for  example,  D  is  —  0.8  from 
OX,  and  C  is  -  1  from  OX  and  also  -  1  from  OY. 

The  two  distances  to  any  point  P  from  OF  and  OX  are  called 
the  rectangular  coordinates  of  P,  and  are  frequently  denoted 


I,  §  9]  INTRODUCTION  1 1 

by  the  letters  x  and  y,  respectively.  The  horizontal  distance  x 
is  called  the  abscissa  of  P;  the  vertical  distance  y  is  called  the 
ordinate  of  P.  In  giving  these  distances  it  is  generally  under- 
stood that  the  first  one  mentioned  is  x,  the  last  y. 

Thus  A,  Fig.  10,  is  briefly  denoted  by  the  numbers  (1.2,  1.4);  B  is 
denoted  by  (-1,  1.2);   C  by  (-1,  -1);  D  by  (1.4,  -0.8). 

The  lines  OX,  0  Fare  called  the  axes  of  coordinates,  or  simply 
the  axes,  OX  is  called  the  aj-axis,  0  Fthe  ^/-axis.  The  point  0 
is  called  the  origin. 

The  four  portions  into  which  the  plane  is  divided  by  the 
axes  are  called  the  first,  second,  third,  and  fourth  quadrants,  as 
in  Fig.  10. 

To  locate  a  point  is  to  describe  its  position  in  the  plane  in 
terms  of  its  distances  from  the  coordinate  axes ;  e.g.  (—5,2)  is 
a  point  5  units  to  the  left  of  the  ^/-axis  and  2  units  above  the 
a>-axis.  To  plot  a  point  is  to  mark  it  in  proper  position  with 
respect  to  a  pair  of  axes. 


EXERCISES  III SQUARED  PAPER 

1.  Locate  and  plot  each  of  the  following  points  with  respect  to  some 
pair  of  axes  : 

(a)  (1,2),    (&)  (2,  -3),    (c)  (4,  -7),  (d)  (-5,2),    (e)   (-7,  -7), 
(/)   (7,  6),    (9)  (5,  12),     (h)  (8,  -3),    (i)   (-6,  -5),   (j)   (6,  -2). 

2.  Show  that  the  line  joining  (5,  —  4)  and  (—  5,  4)  is  bisected  by  the 
origin. 

3.  On  what  lines  do  all  points  (1,  0),  (2,  0),  (-3,  0),  (1.5,  0)  lie? 
On  what  line  do  all  the  points  (0,  0),  (0,  1),  (0,  2),  (0,  5),  (0,  -  2)  lie  ? 
Make  a  general  statement  about  such  points. 

4.  Find  the  distance  from  the  origin  to  each  of  the  points  in  Ex.  1, 
by  using  the  folded  edge  of  another  piece  of  squared  paper. 

Compute  the  same  distances  by  regarding  each  of  them  as  the  length 

of  the  hypotenuse  of  a  right  triangle,  the  lengths  of  whose  sides  can  be 

read  directly  from  the  figure.     Each  of  these  methods  can  be  used  as  a 

check  on  the  other.     Ans.  (a)  2.2,    (6)  3.6,    (c)  8.1,    (d)  5.4,    (e)  9.9, 

(/)  8.6,    (g)  13,  (h)  8.5,     (i)  7.1,    (j)  6.3 

5.  Construct  the  triangle  whose  vertices  are  (6,  2),  (8,  4),  and  (10, 12). 
Find  its  perimeter  and  its  area.  Ans.  21.8,  6. 


12  PLANE  TRIGONOMETRY  [I,  §  9 

6.  Find  the  lengtlis  of  the  segments  whose  end  points  are  :  (a)  (2,  4) 
and  (5,  8)  ;  (6)   (4,  -3)  and  (-1,  3);  (c)   (1,  -2)  and  (4,  2). 

Ans.  5,  7.8,  5. 

7.  Find  the  sides  and  diagonals  of  the  parallelogram  whose  vertices  are 
(2,  1),  (5,  4),  (4,  7),  and  (1,  4).  Ans.  SV%  VlO,  2VT0,  4. 

8.  Plot  the  points  A  :  (1,  0),  JB  :  (-  3,  2),  C :  (1,  1),  D  :  (7,  3)  and 
determine  the  angle  at  which  the  line  AB  crosses  the  line  CD.     Ans.  46°. 

9.  Plot  A:  (2,  1),  5:  (6,  -1),  C:  (1,  3),  D:  (-2,  -3)  and  find 
the  angle  at  which  AB  crosses  CD  ;  also  find  the  area  of  the  triangle  whose 
sides  are  AB,  CD,  and  BD.  Ans.  90°,  16.8 

10.  Plot^:  (5,  -2),  5:  (14,  8),  (7:  (2,  3)  and  find  the  distance  from 
A  to  BC  ;  also  find  the  area  of  the  triangle  ABC.        Ans.  75/13,  37.5 

11.  A  farm  is  described  in  the  deed  as  N.E.  J  and  E.  \  of  N.  W.  J, 
Section  5,  Wayne  Township,  Tippecanoe  County,  Ind.  Taking  the  center 
lines  of  this  section  as  axes,  make  a  map  from  the  following  data :  A 
ditch  crosses  the  farm  through  the  points  (—80,  40),  (80,  80),  (160,  136), 
distances  being  measured  in  rods.  The  house  is  at  (152,  72).  There  are 
seven  fields  whose  corners  are  :  A,  (-  80,  112),  (-  80,  160),  (—  16,  112), 
(-16,  160);  5,  (-80,  40),  (-16,56),  (-16,  112),  (-80,  112); 
G,  (-80,  0),  (0,  0),  (0,  60),  (-80,  40);  D,  (-16,  56),  (80,  80), 
(80,  160),  (-16,  160)  ;  E,  (80,  80),  (160,  136),  (160,  160),  (80,  160); 
F,  (80,  0),  (160,  0),  (160,  136),  (80,  80)  ;  G,  (0,  0),  (80,  0),  (80,  80), 
(0,  60).     Find  the  area  of  each  field  and  the  total  length  of  fence. 

Ans.  19.2,  25.6,  25,  55.2,  26,  54,  35,  (acres);  3  miles  68  rods. 

12.  Positions  on  a  rectangular  farm  are  given  by  their  coordinates  in 
rods,  referred  to  two  sides  of  the  farm  as  axes,  as  follows  :  house  (10,  4), 
barn  (6,  4),  gate  of  pasture  (60,  20).  A  railroad  passes  between  the  house 
and  barn,  with  a  crossing  at  the  point  (3,  12).  Draw  a  map  showing  these 
objects.  Determine  how  much  farther  it  is  from  the  house  to  the  barn  by 
way  of  the  crossing  than  along  the  straight  line  connecting  them.  How 
much  farther  is  it  from  the  barn  to  the  pasture  gate  by  way  of  the  crossing 
than  along  a  straight  line  ?  Ans.  15.2,  9.78 

13.  A  certain  city  park  is  bounded  by  a  main  street,  two  cross  streets 
perpendicular  to  it,  and  a  stream.  The  distances,  in  feet,  to  the  stream 
measured  perpendicularly  from  the  main  street  at  100  ft.  intervals  are 
found  to  be  680,  650,  525,  450,  450,  460,  540.  Draw  a  map  of  the  park  and 
determine  approximately  its  area.  Ans.  7  acres,  9580  sq.  ft. 

14.  To  determine  the  height  of  a  tree  OA  standing  in  a  level  field  the 
distance  OB  =  100  ft.  from  the  base  0  of  the  tree  to  a  point  B  in  the 
field,  and  the  angle  of  elevation  OB  A  =  37°,  are  measured.  Find  the 
height  of  the  tree.  Ans.  75  ft. 


CHAPTER   II 

DEFINITIONS.     SOLUTION   OF  RIGHT  TRIANGLES 

10.  Tables.  While  the  methods  for  solving  triangles  ex- 
plained in  Chapter  I  are  sufficient  for  all  cases,  they  are  really 
not  convenient  where  great  accuracy  is  desired,  since  for  this 
purpose  the  figure  would  need  to  be  drawn  on  a  very  large 
scale.  The  method  usually  employed  when  one  desires  greater 
accuracy  than  can  be  conveniently  attained  by  the  method  of 
construction  and  measurement  is  the  method  of  tables. 
Tables  are  constructed  which  give  approximately  the  ratios 
of  each  pair  of  sides  for  all  right  triangles.  To  obtain  the 
ratio  of  a  certain  pair  of  sides  of  a  right  triangle  with  a  given 
acute  angle  it  is  then  only  necessary  to  consult  the  table. 

For  example,  it  is  known  by  geometry  that  if  one 
angle  of  a  right  triangle  is  30°,  the  side  opposite  this 
angle  is  one-half  the  hypotenuse.  Hence  if  the  hy- 
potenuse is  given,  that  side,  and  hence  also  the  other 
one,  can  be  determined.  If  in  Fig.  11,  AB  =  22.5,  and 
ZA  =  30°,  then  the  side  ^O  =  (1/2) (22.5)=  11.25 

If,  for  an  acute  angle  of  every  right  triangle,  the  ratio  of  the 
opposite  side  to  the  hypotenuse  were  known  to  us,  then  we 
could  solve  every  right  triangle  in  the  same  manner. 

It  will  be  shown  later  that  all  oblique  triangles  can  be  cut 
up  into  right  triangles  in  such  a  way  that  the  same  tables  can 
be  used  in  all  cases  for  solving  oblique  triangles. 

Since  any  triangle  can  be  enlarged  (or  reduced)  in  size  by 
drawing  it  on  a  larger  (or  smaller)  scale,  only  the  ratios  of  the 
sides  are  really  important. 

11.  Definitions  of  the  Ratios.  As  indicated  in  §  10,  the 
ratio  of  two  sides  of  a  triangle  does  not  depend  upon  the  size 

13 


14 


PLANE  TRIGONOMETRY 


[II,  §  11 


of  the  triangle,  but  only  upon  the  angles.  Thus  in  the  right 
triangles  MPN,  MP'N',  MP^'N"  of  Fig.  12,  in  which  PiV, 
P'N',  P'^N''  are  perpendicular  to 
MN,  the  ratios  NP/MP,  N'P'/MP\ 
N''P  "IMP  "  are  all  equal.  Moreover,  if 
piti^ni  ig  dra^n  perpendicular  to  MP^ 
each  of  the  ratios  just  mentioned  is  equal 
to  N'"P"'/MP"\  (Why?)  These  ra- 
tios, then,  depend  only  on  the  angle  a  at  M.  It  is  convenient  to 
place  the  angle  on  a  pair  of  axes  so  that  the  vertex  falls  at  the 
origin  0,  one  side  lies  along  the  a^axis,  to  the  right,  and  the 
other  side  falls  in  the  first  quadrant.  On  this  side  take  any 
point  P  at  random,  except  0,  and  drop 
the  perpendicular  PM  to  the  a?-axis 
(see  Fig.  13).  Let  OP=r)  then  by 
geometry 

r  =  -Vx^  +  2/2,* 

where  x  and  y  are  the  coordinates  of 

the  point  P.     The   various   ratios    of 

pairs  of  the  three  quantities  x,  y,  r  are  the  same  for  all  points  P 

taken  in  the  side  OP  of  the  angle  a.     These  are : 

(1) 

(2) 
(3) 


Fig.  13. 


y 


,  called  the  sine  of  the  angle  a,  written  sin  a. 
-,  called  the  cosine  of  the  angle  a,  written  cos  a. 
— ,  called  the  tangent  of  the  angle  a,  written  tan  a. 


The  reciprocals  f  of  these  ratios  are  also  often  used  ; 

(4)  r/y  is  called  the  cosecant  of  the  angle  a,  written  esc  a. 

(5)  r/x  is  called  the  secant  of  the  angle  a,  written  sec  a. 

(6)  x/y  is  called  the  cotangent  of  the  angle  a,  written  ctn  a. 


*  The  radical  sign  is  used  to  denote  the  positive  square  root. 

t  The  reciprocal  of  a  number  is  unity  divided  by  the  number.  The  recipro- 
cal of  a  common  fraction  is  the  result  of  inverting  it ;  thus  the  reciprocal  of 
y/r  is  r/y.    Every  number  has  a  reciprocal  except  0,  which  has  not. 


II,  §  13]  DEFINITIONS  15 

These  six  ratios  are  collectively  called  trigonometric  ratios 
or  also  trigonometric  functions  of  the  angle. 

Other  expressions  derived  from  these  are  also  frequently  used  ;  for  ex- 
ample, many  engineers  use  the  following  combinations  : 

(7)  versed  sine  of  a  =  1  —  cos  a,  written  vers  a  ; 

(8)  external  secant  of  a  =  sec  a  —  1,  written  exsec  a  ; 

(9)  haversine  of  a  =  half  the  versed  sine  of  a 

=  ^-^'^^''^  written  hav  a  ; 

2 

and  occasionally  also  the  function  coversed  sine  of  a  =  1  —  sin  a,  written 
covers  a. 

12.  Right  Triangles.  In  the  right  triangle  OFM,  Fig.  13,  y 
is  the  side  opposite  the  angle  a,  x  is  the  side  adjacent  to  a, 
and  r  is  the  hypotenuse.  From  the  definitions  (l)-(3),  we  see 
that  in  any  right  triangle : 

(10)  The  sine   of  either  acute  angle  =  — — ;    \ 

^    ^  ^  ^  hypotenuse  '     \ 

side  adjacent       \ 


(11)   The  cosine  of  either  acute  angle 


hypotenuse 


(12)  The  tangent  of  either  acute  angle  =  -r-z — ^? -;  / 

and,  after  clearing  of  fractions,  we  find  for  either  acute  angle 

(13)  The  side  opposite  =  hypotenuse  x  sine 

=  side  adjacent  x  tangent; 

(14)  The  side  adjacent  =  hypotenuse  x  cosine 

=  side  opposite  x  cotangent; 

/-I  i-\    TT  ^  ^  side  opposite      side  adjacent 

(15)  Hypotenuse  = f^ = r • 

^    ^  sine  cosine 

The  student  should  so  thoroughly  learn  these  statements 
that  he  can  apply  them  instantly  and  confidently  to  any  right 
triangle  that  he  sees,  whatever  its  position  in  the  plane. 

13.  Elementary  Relations.  The  trigonometric  functions 
are  connected  by  many  simple  relations.     Thus  : 

,^  />x  ^  sin  a      .         y     y     ^ 

(Id)  tan  a  = ,  since      =  -  -; — 

cos  ot  X     r     r 


16 


PLANE  TRIGONOMETRY 


[11,  §  13 


Similarly,  the  student  can  easily  show  that 
sm  a         1 


(17) 

(18)    sec  a  = 


ctn  a  = 


cos  a 


cos  a      tana' 

(19)    CSC  a 


1 

sin  a 


.^ -2 


Other  relations  will  be  given  later. 

The  following  examples  illustrate  a  method  of  constructing 
an  angle  when  one  of  its  ratios  is  given. 

Example  1.     Construct  an  acute  angle  whose  sine  is  2/7. 

To  construct  such  an  angle  draw  a 
right  triangle  whose  hypotenuse  is  7 
and  one  whose  side  is  2.  This  can 
easily  be  done  on  cross-section  paper. 
With  a  radius  of  7  draw  a  circle  and 
mark  its  intersection  with  the  hori- 
zontal ruling  2  units  above  the  center. 
The  angle  between  the  horizontal 
diameter  and  the  radius  to  this  intersection  is  the  angle  required. 
Example  2.  Construct  an  acute  angle  whose  tangent  is  3/8. 
This  is  most  easily  done  by 
drawing  a  triangle  whose  base  is 
8  and  whose  altitude  is  3.  The 
angle  between  the  hypotenuse  and 
base  is  the  angle  required.  As  in 
Example  1,  it  will  be  found  conve- 
nient to  draw  the  figure  on  cross- 
section  paper.  Fig.  15. 


Fig.  14. 


3 

8  \ 


EXERCISES   IV.  —  TRIGONOMETRIC   RATIOS 

1.  On  cross-section  paper  construct  angles  whose  sines  are  :•  (a)  1/5; 
(6)  2/5;  (c)  3/5;  (d)  4/5;  (e)  2/3;  (/)  5/7;  {g)  0.5 

2.  Is  there  an  acute  angle  whose  sine  is  any  given  positive  number  ? 

3.  Construct  angles  whose  tangents  are :   (a)  3/10;  (6)   1/2;  (c)  2/3; 
(d)  1;   (6)   10/3;   (/)  2;   {g)  7.5;  (A)  3.4;  (i)  1.7 

4.  Is  there  an  acute  angle  whose  tangent  is  any  given  number  ? 

6.    How  large,  in  degrees,  is  the  acute  angle  whose  tangent  is  1  ? 

6.   How  does  the  angle  whose  tangent  is  2  compare  with  the  angle 
whose  tangent  is  1  ?    Check  your  answer  by  drawing  an  accurate  figure. 


11,  §  13] 


DEFINITIONS 


17 


14.   Construction  of  Small  Tables.     Approximate  values  of 
the  trigonometric  functions  of  a  given  acute   angle  may  be 


20  20  30  iO  60  HO  70  iO  itO  100 

Fig.  16. 
found   by   measurement  as    follows.     On  a  sheet  of  squared 
paper,  construct  a  quarter  circle  with  its   radius  =  100,  and 


18 


PLANE  TRIGONOMETRY 


[11,  §  14 


with  its  center  at  the  intersection  of  two  heavy  rulings. 
Draw  a  tangent  to  this  circle  perpendicular  to  the  horizontal 
rulings.  Given  now  any  acute  angle,  a,  lay  it  off  above  the 
horizontal  axis  with  its  vertex  at  the  center  of  the  circle. 
Call  the  points  where  its  side  crosses  the  circle  and  the  tan- 
gent P  and  Q,  respectively.  Then  the  ordinate  (y)  of  the 
point  P  can  be  read  at  least  to  units,  and  this  divided  by 
r  =  100  gives  the  value  of  sin  a  to  two  decimal  places. 
Similarly,  the  abscissa  (x)  of  P  can  be  read  to  units,  and  this 
divided  by  100  gives  cos  a.  Likewise  the  ordinate  of  Q  can 
be  read  to  units,  and  this  divided  by  100  gives  tan  a. 
Finally,  ctn  a,  sec  a,  esc  oc,  can  be  computed  as  the  reciprocals 
of  tan  a,  cos  a,  sin  a,  respectively.  The  student  will  find  it 
instructive  to  compute  in  this  way,  from  Fig,  16,  values  to  fill 
out  the  following  table. 


a 

5'' 

10° 

15" 

20° 

25° 

30° 

35° 

40° 

45° 

50° 

55° 

60° 

65° 

70° 

75° 

80° 

85° 

since 

cos  a 





tan  a 

etna 

15.  Functions  of  Complementary  Angles.  If  all  of  this 
table  is  filled  out  correctly,  it  will  be  found 
that  every  number  in  it  occurs  twice ;  once  for 
an  angle  less  than  45°  and  once  for  an  angle 
greater  than  45°.  This  result  indicates  that 
the  sine  of  any  angle  is  the  cosine  of  its  com- 
plement; and  the  tangent  of  any  angle  is  the  cotangent  of  its 
complement. 

These  relations  will  now  be  proved  for  any  acute  angle  a. 
Let  p  =  90°  —  a ;  then  a  and  fi  are  the  acute  angles  of  a 
right  triangle.     Denote  the  sides  opposite  a  and  fihj  a  and  b. 


Fig.  17. 


II,  §  16]        SOLUTION  OF  RIGHT  TRIANGLES  19 

respectively  j  and  the  hypotenuse  by  c.     Then  by  §  12, 

side  opposite      a 

sm  a  =  -r f^ =  - ; 

hypotenuse       c 

^  __  side  adjacent  __  a 

^  ""    hypotenuse       c  ' 

side  opposite  __a 

"~  side  adjacent     b  ' 

^     _       side  adjacent     a 

ctn  fi  =  -r^ ^ — —  =-  ; 

side  opposite     b  ' 

whence,  remembering  that  /3  =  90°  —  a, 

(20)  sin  a  =  cos  )8  =  cos  (90°  -  a), 

(21)  tan  a  =  ctn  /?  =  ctn  (90°  -  a). 
In  the  same  way  it  can  be  shown  that 

(22)  sec  a  =  esc  (90°  -  a). 

16.  Applications.  The  values  of  the  trigonometric  ratios 
have  been  computed  approximately  for  all  acute  angles,  and 
recorded  in  convenient  tables.  These  tables,  together  with  the 
formulas  just  given,  enable  us  to  solve  all  cases  of  right 
triangles.  On  page  21  is  printed  a  table  giving  the  values  of 
the  ratios  to  three  decimal  places.  If  still  greater  accuracy  is 
required,  a  four  or  a  five-place  table  should  be  employed.  In 
the  following  examples  the  three-place  table  is  used. 

Example  1 .  One  angle  of  a  right  triangle  is  38°  and  the  hypotenuse 
is  12  ft.     Find  the  lengths  of  each  of  the  other  sides. 

Draw  a  figure,  mark  the  given  parts,  and  indicate 
the  parts  to  be  found  by  suitable  letters,  say  x  and  y. 
The  sides  x  and  y  are  then  respectively  the  side  ad- 
jacent and  the  side  opposite.  To  find  «,  note  that 
the  hypotenuse  is  given  ;  hence  by  (14),  §  12, 

X  =  12  .  cos  38°. 

The  value  of  the  cosine  of  38°  from  the  three  place  table  is  found  to  be  .788 
Using  this  value  we  find 

x=:12  (.788) 
.788 

12 

or  X-       9.456 


20 


PLANE  TRIGONOMETRY 


[11,  §16 


Example  2. 


Fig.  19. 


Similarly  by  equation  (13) ,  §  12, 

2/  =  12  .  sill  38^ 
and  from  the  three-place  table  the  sine  of  38°  is  found  to  be  .616.    Using 
this  value  we  obtain  y  =  12  (.616) 

.616 

12 

y=       7.392 

As  a  check,  the  Pythagorean  theorem  may  be  used,  particularly  if  a 
table  of  squares  is  available.     Thus,  denoting  the  hypotenuse  by  h,  we 

should  have  

h  =  V(9.456)2  +  (7.392)2  =  12.002 

This  agrees  reasonably  well  with  the  given  value  h  =  12.  Another  check 
that  is  more  practical  is  given  by  measurement  from  a  good  figure. 

One  side  of  a  right  triangle  is  17  and  the  angle  opposite 
this  side  is  27°  ;  what  is  the  length  of  the  hypote- 
nuse ?  of  the  other  side  ? 

Denote  the  hypotenuse  by  u  and  the  unknown 
side  by  v.  Noting  that  the  side  opposite  the  given 
angle  is  given,  find  the  side  adjacent^  v,  by  (14),  *§  12. 
To  find  the  hypotenuse,  use  (15),  §  12  : 

v  =  17.  ctn  27°  =  17(1.963) 
1.963 

17 

13.741 

19.63 
V  =      33.371 
w  =  17  -  sin  27°  =  17  ^.464 

Performing  the  division  we  find 

'    u  =  37.44 

Check  these  answers  by  drawing  an  accurate  figure. 

Example  3.    The  hypotenuse  of  a  right  triangle  is  41  and  one  side  is 
13  ;  find  the  opposite  angle.  ^ ^^  ;. 

Denote  the  opposite  angle  by  a,  then  by 
equation  (10),  §  12, 

sin  a  =  13  -f-  41  =  .317 

From  the  table  (p.  21)  we  see  that  sin  18°  =  .309  and  that  sin  19°  =  .326, 
so  that  sin  a  is  very  nearly  halfway  between  sin  18°  and  sin  19°.  We 
judge  therefore  that  the  angle  a  is  about  halfway  between  18°  and  19°; 
hence  a  =  18°  .5 


II,  §  16]        SOLUTION  OF  RIGHT  TRIANGLES 


21 


TRIGONOMETRIC  FUNCTIONS   TO  THREE  PLACES   OP  DECIMALS 


a 

sin  a 

sec  a 

tan  a 

ctn  a. 

CSC  a 

cos  a 

0° 

.000 

1.000 

.000 

1.000 

90° 

1° 

.017 

1.000 

.017 

57.290 

57.299 

1.000 

89° 

2° 

.035 

1.001 

.035 

28.636 

28.654 

.999 

88° 

3° 

.052 

1.001 

.052 

19.081 

19.107 

.999 

87° 

40 

.070 

1.002 

.070 

14.301 

14.336 

.998 

86° 

5^ 

.087 

1.004 

.087 

11.430 

11.474 

.996 

85° 

6° 

.105 

1.006 

.105 

9.514 

9.567 

.995 

84°  . 

70 

.122 

1.008 

.123 

8.144 

8.206 

.993 

83° 

8^ 

.139 

1.010 

.141 

7.115 

7.185 

.990 

82° 

9° 

.156 

1.012 

.158 

6.314 

6.392 

.988 

81° 

10° 

.174 

1.015 

.176 

5.671 

5.759 

.985 

80° 

11° 

.191 

1.019 

.194 

5.145 

5.241 

.982 

79° 

12° 

.208 

1.022 

.213 

4.705 

4.810 

.978 

78° 

13° 

.225 

1.026 

.231 

4.331 

4.445 

.974 

77° 

14° 

.242 

1.031 

.249 

4.011 

4.134 

.970 

76° 

15° 

.259 

1.035 

.268 

3.732 

3.864 

.966 

75° 

16° 

.276 

1.010 

.287 

3.487 

3.628 

.961 

74° 

17° 

.292 

1.046 

.306 

3.271 

3.420 

.956 

73° 

18° 

.309 

1.051 

.325 

3.078 

3.236 

.951 

72° 

19° 

.326 

1.058 

.344 

2.904 

3.072 

.946 

71° 

20° 

.342 

1.064 

.364 

2.747 

2.924 

.940 

70° 

21° 

.358 

1.071 

.384 

2.605 

2.790 

.934 

69° 

22° 

.375 

1.079 

.404 

2.475 

2.669 

.927 

68° 

23° 

.391 

1.086 

.424 

2.356 

2.559 

.921 

67° 

24° 

.407 

1.095 

.445 

2.246 

2.459 

.914 

66° 

25° 

.423 

1.103 

.466 

2.145 

2.366 

.906 

65° 

26° 

.438 

1.113 

.488 

2.050 

2.281 

.899 

64° 

27° 

.454 

1.122 

.510 

1.963 

2.203 

.891 

63° 

28° 

.469 

1.133 

.532 

1.881 

2.130 

.883 

62° 

29° 

.485 

1.143 

.554 

1.804 

2.063 

.875 

61° 

30° 

.500 

1.155 

.577 

1.732 

2.000 

.866 

60° 

31° 

.515 

1.167 

.601 

1.664 

1.942 

.857 

59° 

32° 

.530 

1.179 

.625 

1.600 

•  1.887 

.848 

58° 

33° 

.545 

1.192 

.649 

1.540 

1.836 

.839 

57° 

34° 

.559 

1.206 

.675 

1.483 

1.788 

.829 

56° 

35° 

.574 

1.221 

.700 

1.428 

1.743 

.819 

66° 

36° 

.588 

1.23() 

.727 

1.376 

1.701 

.809 

54° 

37° 

.602 

1.252 

.754 

1.327 

1.662 

.799 

53° 

38° 

.616 

1.269 

.781 

1.280 

1.624 

.788 

52° 

39° 

.629 

1.287 

.810 

1.235 

1.589 

.777 

51° 

40° 

.643 

1.305 

.839 

1.192 

1.556 

.766 

60° 

41° 

.656 

1.325 

.869 

1.150 

1.524 

.755 

49° 

42° 

.669 

1.346 

.900 

1.111 

1.494 

.743 

48° 

43° 

.682 

1.367 

.933 

1.072 

1.466 

.731 

47° 

44° 

.695 

1.390 

.966 

1.036 

1.440 

.719 

46° 

46° 

.707 

1.414 

.1000 

1.000 

1.414 

.707 

46° 

COS  a 

CSC  a 

ctn  a 

tana 

sec  a 

sin  a 

a 

22  PLANE  TRIGONOMETRY  [II,  §16 

Example  4.  The  two  perpendicular  sides  of  a  right  triangle  are  23 
and  83  ;  determine  the  acute  angles  and  the  hypotenuse. 

Denote  the  hypotenuse  by  h  and  the  angle  opposite  the  smaller  side 
by  a  ;  then  by  equation  (12)  §  12, 

tan  a  =  23  -r-  83. 
After  performing  the  division  it  is  found  that 

tan  a  =  .277 
As  in  the  example  above  it  is  noticed  that  tan  a  lies  very  nearly  halfway 
between  tan  15°  and  tan  16°  ;  we  have,  therefore,  very  approximately, 

a  =  15°.5 

17.  Directions  for  Solving  Triangles.  In  the  solution  of 
triangles,  use  the  following  procedure : 

(a)  Draw  a  diagram  approximately  to  scale,  indicating  the 
given  parts.  Mark  the  unknown  parts  by  suitable  letters,  and 
estimate  their  values. 

(6)  If  one  of  the  given  parts  is  an  acute  angle y  consider  the  re- 
lation of  the  known  parts  to  the  one  which  it  is  desired  to  find, 
and  apply  the  proper  one  of  formulas  (10)  •••  (15),  §  12. 

(c)  If  two  sides  are  given,  and  one  of  the  acute  angles  is 
desired,  think  of  the  definition  of  that  function  of  the  angle 
which  employs  the  two  given  sides. 

(d)  Check  each  result. 

EXERCISES  v.  — SOLUTION  OF  RIGHT  TRIANGLES 

1.  One  side  of  a  right  triangle  is  21  ;  the  adjacent  angle  is  42°  ;  de- 
termine the  remaining  side  and  the  hypotenuse.     Check. 

2.  One  side  of  a  right  triangle  is  21  and  the  opposite  angle  is  42°  ;  de- 
termine the  remaining  side  and  hypotenuse.     Check. 

3.  The  hypotenuse  of  a  right  triangle  is  28  ;  one  angle  is  32°.  Deter- 
mine the  two  perpendicular  sides.     Check. 

4.  What  is  the  angle  of  inclination  of  a  roof  which  has  half  pitch  ? 
1/3  pitch? 

[Note.  The  pitch  of  a  roof  is  equal  to  the  height  of  the  comb  above 
the  eaves  divided  by  the  total  distance  between  the  eaves.  ] 

5.  In  the  following  triangles  h  denotes  the  hypotenuse  ;  the  angle  A 
is  opposite  the  side  a  and  the  angle  B  is  opposite  the  side  b.  Use  the 
table  to  compute  the  unknown  parts  from  the  given  parts.     Check. 


II,  §19]  SOLUTION  OF  RIGHT  TRIANGLES  23 

(a)  A  =  61^  b  =  41.  (d)  A  =  32°,  a  =  330. 

(6)    a  =  421,  6  =  401.  (e)    a  =  313,  h  =  720. 

(c)     a  =  62,    /I  =  125.  (/)  B  =  49°,  h  =  24. 

6.  Determine  the  height  of  a  tower  MN,  if  the 
horizontal  distance  EM  to  it  is  450  ft.  and  the  angle  of 
elevation  MEN  is  27°.     Check. 

7.  A  vertical  pole  35  ft.  high  casts  a  horizontal    ^  •^^^^ 
shadow  45  ft.  long.     Determine  the  angle  of  elevation 
of  the  sun  above  the  horizon.     Check. 

8.  An  object  known  to  be  100  ft.  in  height  stands  on  the  bank  of  a 
river;  from  the  opposite  bank  of  the  river  the  angle  of  elevation  of  the  top 
of  the  object  is  found  to  be  24°;  find  the  width  of  the  river.     Check. 

9.  The  radius  of  a  circle  is  7  ft.  What  angle  will  a  chord  of  the  circle 
11  ft.  long  subtend  at  the  center  ?     Check. 

10.  From  the  top  of  a  cliff  92  ft.  in  height  the  angle  of  depression  of  a 
boat  at  sea  is  observed  to  be  20°.     How  far  out  is  the  boat  ?    Check. 

11.  To  find  the  distance  between  two  objects  A  and  B,  where  5  is  in  a 
swamp,  the  distance  AG  =  350  ft.  is  measured  at  right  angles  to  the  fine 
joining  them.  At  G  an  observer  holds  an  ordinary  rake  with  the  end  of 
the  handle  at  his  eye  and  with  the  center  of  the  rake  directed  toward  A. 
There  appear  then  to  be  6  teeth  of  the  rake  between  A  and  B.  If  the 
teeth  are  one  inch  apart  and  the  handle  of  the  rake  is  five  feet  long,  de- 
termine the  distance  between  A  and  B. 

18.  The  Question  of  Greater  Accuracy.  The  degree  of 
accuracy  of  the  results  obtained  by  using  the  values  of  the 
trigonometric  functions  to  three  places  of  decimals,  while 
sufficient  for  many  ordinary  applications,  is  not  satisfactory 
for  some  purposes ;  for  example,  in  extended  surveys,  in 
astronomy,  and  in  any  work  for  which  the  data  must  be  deter- 
mined by  using  instruments  of  precision. 

More  accurate  values  have  been  calculated.  The  values  for 
angles  at  intervals  of  V  are  given  to  five  decimal  places  in  five- 
place  tables.* 


*  Throughout  this  book,  page  references  to  Tables  are  to  The  Macmillan 
.Tables.  These  tables  may  be  had  separately  bound.  They  are  bound  with 
this  book  in  the  edition  with  complete  tables.  The  edition  of  this  book  with 
brief  tables  contains  only  four-place  tables,  for  the  convenience  of  those  who 
prefer  the  full  tables  separately  bound. 


24 


PLANE  TRIGONOMETRY 


[II,  §19 


19.  Use  of  the  Large  Tables.  Five-place  tables  are  used  in 
precisely  the  same  manner  as  the  small  table  of  p.  21. 

Example  1 .  One  angle  of  a  right  triangle  is  42°  20'  and  the  hypotenuse 
is  28  ft.  6  in.  long.  Find  the  remaining  sides  and  the  other  angle.  Draw 
a  diagram  to  illustrate  the  problem,  indicating  the  given  parts.  Denote 
the  unknown  parts  by  the  letters  a  and  6,  as  in  Fig.  22. 

To  find  6,  note  that  it  is  the  side  adjacent  to  the  given  angle,  and  that 
the  hypotenuse  is  given.     Hence,  by  (14),  §  12, 
b  =  28.5  cos  42^20'  =  28.5  x  .73924  =  21.07 
Note  that  a  is  opposite  the  given  angle;  hence 
by  (13),  §  12. 

a  =  28.5  sin  42°  20'  =  28.5  x  .67344  =  19.19 
the  sine  and  the  cosine  of  42°  20'  being  found  in 
the  Tables,  p.  43. 
The  angle  /3,  being  the  complement  of  42°  20',  is  47°  40'. 
Example  2.    .The  perpendicular  sides  of  a  right  triangle  are  22  ft.  6  in. 
and  54  ft.^  respectively.     Find  the  hypotenuse  and  the  angles. 

Draw  a  diagram,  indicating  the  given  parts  and  lettering  the  parts  to 
be  found,  as  in  Fig.  23.  To  find  a,  note  that  the  given  parts  are  the  sides 
opposite  and  adjacent  to  it  ;  hence  by  the  definition  of  tangent,  we  write 

tan  a  =  22.5  --  54  =  .41667 
From  the  Tables,  p.  33, 

tan  22°  37'.  =  .41660  and  tan  22°  38'  --=  .41694 
whence 

a  =  22°  37'+  and  /3,  its  complement,  is  67°  23'-. 
By  the  Pythagorean  theorem  of  plane  geometry,  using 
a  table  of  squares  and  square  roots,  Tables  p.  94, 

/i2  =  54^  +  22^5^  =  3422.25 
whence,  /t  =  58.5  Tables,  p.  103. 

Another  method  of  finding  h  is  the  following:  Having 
found  a  =  22"  SI',  h  =  54/cos  22°  37'  =  54/.92310  =  58.498  by  (15)  §  12. 
However,  this  method  is  open  to  the  objection  that  any  error  made  in 
computing  a  vitiates  the  resulting  value  found  for  h.  In  general,  com- 
pute each  unknown  part  from  the  given  parts ;  i.e.  do  not  use  computed 
parts  as  data  if  it  can  he  avoided. 

In  solving  right  triangles,  observe  carefully  the  directions  of 
§  17,  p.  22,  and  use  five-place  values  of  the  functions  (Tables, 
pp.  22-44  and  pp.  94-111)  as  illustrated  in  the  preceding 
examples. 


II,  §19]         SOLUTION  OF  RIGHT  TRIANGLES 


25 


EXERCISES  VI.  — RIGHT  TRIANGLES 

1.  Solve  the  following  right  triangles.  The  hypotenuse  is  denoted  by 
/i,  other  sides  by  other  small  letters,  and  any  angle  by  the  capital  letter 
corresponding  to  the  small  letter  that  denotes  the  side  opposite  it. 

(a)  A  =  61°  17',  b  =  1.4      (d)  M=  49°  49',  /t=24.6      (g)  p  =  lS.2,  g  =  50. 
(6)  A  =  32°  31',  a  =  33.      (e)  b  =  4.848,  h  =  10.      (h)  u  =  11.65,  h=2^, 

(c)  A  =  62.12,  h  =  254.      (/)  C7=  63°  2',  u  =  40.     (i)  m=34.2,  h  =100. 
Ans.    (a)  2.56,2.91;    (6)  51.77,  61.39;    (c)    14°  9'.4,  75°  50'. 7,  246.29  ; 

(d)  18.80,  15.87  ;  (e)  61°,  29°,  8.746  ;  (/)  20.35,  44.88  ;  (g)  20°+,  70°-, 
53.21  ;  (h)  27°  46'.5,  62°  13'.5,  22.12  ;  (Q   70°,  20°,  93.97 

2.  In  the  following  right  triangles  find  the  side  not  given  : 


(a) 

(h) 

(c) 

(d) 

(e) 

(f) 

(9) 

(h) 

(0 

(J) 

(k) 

G) 

side 

2.19 

45.6 

5.82 

53.4 

73.6 

25.6 

46 

17.5 

46.5 

6.83 

13.5 

106 

^JV. 

7.75 

9.43 

54.4 

45.5 

9.92 

35.1 

535.3 

side 

82.5 

19.2 

138 

110.4 

42.7 

ans. 

7.43 

94.26 

7.42 

56.75 

156.4 

48 

119.6 

42 

63.13 

7.19 

32.4 

524.7 

3.  In  each  of  the  following  right  triangles  find  the  three  parts  not  given 
and  the  area. 

(a)  a  =  30.2,  h  =  33.3  Ans.  24°  55' .1,  65°  4'.9,  14.03,  211.85 

(b)  A  =  35°,  b  =  100.  Ans.  70.021,  122.07,  3501. 

(c)  h  =  4S,  B  =  27°.  Ans.  19.52,  38.31,  373.98 

(d)  h:=z  176,  A  =  32°.  ^ns.  93.26,  149.25,  6959.68 

(e)  /i  =  425,  6  =  304.  Ans.  45°  40',  297,  45144. 

4.  The  base  of  an  isosceles  triangle  is  324  ft.,  the  angle  at  the  vertex  is 
64°  40'.     Find  the  equal  sides  and  the  altitude.  Ans.  302.89,  255.93 

5.  The  shadow  of  a  tower  200  ft.  high  is  252.5  ft.  long.  What  is  the 
angle  of  elevation  of  the  sun  ?  Ans.  38°  23'. 

6.  A  chord  of  a  circle  is  21.5  ft.,  the  angle  which  it  subtends  at  the 
center  is  41°.     Find  the  radius  of  the  circle.  Ans.  30.7 

7.  To  determine  the  width  AB  of  a  river,  a  line  BC  100  rods  long  is 
laid  off  at  right  angles  to  a  line  from  B  to  some  object  A  on  the  opposite 
bank  visible  from  B.     The  angle  BCA  is  found  to  be  43°  35^     Find  AB. 

Ans.  95.17 

8.  What  is  the  angle  of  elevation  of  a  mountain  slope  which  rises 
238  ft.  in  one-eighth  of  a  mile  (up  the  slope)?  Ans.  21°  8'+. 


26  PLANE  TRIGONOMETRY  [II,  §19 

9.  Two  ships  in  a  vertical  plane  with  a  lighthouse  are  observed  from 
its  top,  which  is  200  ft.  above  sea  level.  The  angles  of  depression  of  the 
two  ships  are  15^^  17'  and  11°  22^    Find  the  distance  between  the  ships. 

Arts.  262.96 

10.  A  flagstaff  stands  on  the  top  of  a  house.  At  a  point  100  ft.  from 
the  house  the  angles  of  elevation  of  the  bottom  and  top  of  the  staff  are 
respectively  21°  60'  and  33°  3'.     Find  the  height  of  the  staff.     Ans,  26. 

11.  A  24-foot  ladder  can  be  so  placed  in  a  street  as  to  reach  a  window 
16  ft.  high  on  one  side  and  by  turning  it  over  on  its  foot  it  will  reach  a 
window  14  ft.  high  on  the  other  side.    Find  the  width  of  the  street. 

Arts.  37.38 

12.  The  length  of  one  side  of  a  regular  pentagon  is  24  ft.  Find  the 
lengths  of  the  radii  of  the  inscribed  and  circumscribed  circles  and  the 
area.  Ans.  16.62,  20.42,  991.2 

13.  The  side  of  a  regular  decagon  is  10  in.  long.  Find  the  radii  of  the 
inscribed  and  circumscribed  circles  and  the  area. 

Ans.  15.39,  16.18,  769.6 

14.  A  round  silo  21.6  feet  in  diameter  subtends  a  horizontal  angle  of 
6°.     Find  the  distance  from  the  observer  to  the  silo.  Ans.  236.7 

15.  In  an  isosceles  right  triangle  show  that  lines  from  either  base 
angle  to  the  points  of  trisection  of  the  opposite  side  cut  off  respectively, 
one-fifth  and  one-half  the  altitude  from  the  hypotenuse  to  the  vertex  of 
the  right  angle. 


CHAPTER   III 


TRIGONOMETRIC   RELATIONS 

20.  Introduction.  A  few  simple  trigonometric  relations 
have  been  given  in  §  §  12,  13,  and  15.  In  this  chapter  we  shall 
obtain  others.  The  student  should  first  review  those  already 
given. 

21.  Pythagorean  Relations.  The  following  equation  be- 
tween the  abscissa  x,  the  ordinate  y,  and  the  radius  r  is  true 
for  every  point  in  the  plane :  ^ 

(1)  x^+y'^  =  r\ 
Dividing  by  r^,  we  obtain 

but  by  §  11,  at  least  when  a  is  acute, 
xjr  =  cos  a,   yjr  =  sin  a ;  hence 

(2)  sin2  a  +  cos2  a  =  1 ; 

^^^e.  tlie,  sum  of  the  squares  of  the  sine  and  cosine  of  any  acute 
angle  is  equal  to  unity. \ 

Dividing  (1)  by  x^,  and  then  by  y'^,  we  obtain  respectively : 

(3)  1  +  tan2  a  =  sec^  «, 

(4)  1  +  ctn2  a  =  csc2  a. 

Formulas  (2),  (3),  and  (4)  are  examples  of  trigonometric 
identities.     An  identity  in  any  quantity,  a,  is  an  equation  con- 


^x 


Fig.  24. 


*  Formulas  (2),  (3),  and  (4)  are  called  the  Pythagorean  relations  because 
they  are  obtained  from  this  equation,  which  is  the  Pythagorean  theorem  of 
plane  geometry. 

t  This  statement,  as  well  as  (3)  and  (4)  below,  will  later  be  found  to  hold 
for  all  angles,  for  the  general  definitions  of  sine  and  cosine. 

27 


28 


PLANE  TRIGONOMETRY 


[HI,  §21 


taining  a  which  is  satisfied  by  every  value  of  a  for  which  both 
members  are  defined.  Many  other  examples  of  identities  will 
be  found  in  the  pages  that  follow. 

These  formulas  and  those  of  §  13  are  often  useful  in  simplify- 
ing expressions  or  in  verifying  equations.  Other  interesting 
relations  are  given  in  exercises  that  follow. 

Example  1.     To  show  that  sin^  a  —  cos*  a  =  sin2  a  —  cos2  a. 

The  expression  on  the  left  is  the  difference  of  two  squares  and  can 

therefore  be  factored  ;  hence  we  have  sin^  a  —  cos*  a  =  (sin2  a  +  cos2  a) 

(sin2  a  —  cos2  a)  which  is  equal  to  sin2  a  —  cos2  a,  since  sin2  a  +  cos2  a  =  1. 

The  formulas  may  also  be  used  to  compute  the  value  of  one  of  the 

trigonometric  functions  from  that  of  another. 

Example  2.     Given  tan  6  =  5/12,  to  find  cos  0. 

Analytic  Method,  By  (3),  1  +  tan2^  =  sec2^  ;  hence,  sec2^  =  1  + 
25/144  =  169/144,  or  sec  d  =  13/12.  Hence,  cos  0  =  12/13,  since  cos  0 
=  l/sec  0. 

Geometric  Method,     The  following  method  is  much  more  practical,  and 

is  easily  applied  to  any  example  of 
this  sort. 

Draw  a  right  triangle  whose  base 
is  12  and  whose  altitude  is  5.  The 
hypotenuse  is  easily  found  to  be  13. 
It  follows  that 


^ 

^ 

^ 

^ 

> 

^ 

5 

^ 

*«^ 

^ 

L^ 

e 

12 

Fia.  25. 


^^^^^sMe^djacent^-^2/13. 
hypotenuse 


EXERCISES  VII.  — PYTHAGOREAN  RELATIONS.    IDENTITIES 

1.  In  exercises  (a)  —  {%)  determine  the  values  of  the  remaining  func- 
tions of  the  acute  angle  0  by  each  of  the  methods  of  Example  2,  above. 

(a)  sin  ^  =  3/5.  (&)  sin  <9  =  1/3.  (c)  cos  ^  =  1/3. 

(d)  sin  0  =  5/13.  (e)  tan  0  =  VS.  (/)  tan  0  =  3/4. 

(gr)  tan  0  =  1/m.  (h)  sin  0  =b/c.  (i)  sec  0  =  2. 

Prove  the  following  relations  for  any  acute  angle  0: 

2.  (sin  ^  +  cos  0)2  zz:  1  +  2  sin  0  cos  0.  3.    cos  0  tan  0  =  sin  0. 
4.  tan  0  +  ctn  ^  =  sec  ^  esc  0.                          5.   sin  ^  sec  ^  =  tan  0. 

6.  (sec  0  —  tan  0)  (sec  0  +  tan  ^)  =  1. 

7.  (sin3  0  +  cos3  0)  z=  (sin  ^  +  cos  ^)  (1  -  sin  0  cos  0). 

8.  cos2  0  -  sin2  ^  =  1  -  2  sin2  0  =  2  cos2  ^  -  1. 

9.  sec2  0  csc2  0  =  tan2  0  -f  ctn2  0  -\- 2. 


Ill,  §23]  TRIGONOMETRIC  RELATIONS     -  29 

22.  Functions  of  0°  and  90°.  If  an  angle  of  0°  be  placed  on 
coordinate  axes  and  the  construction  of  page  14  be  made,  the 
point  P  will  lie  on  the  ic-axis,  and  we  shall  have 

x=ry  y  =  0. 

The  functions  sine,  cosine,  tangent,  and  secant  of  0°  are 
defined  by  the  same  ratios  as  are  the  corresponding  functions 
of  acute  angles  :  hence  as  in  (1),  (2),  (3),  and  (5),  page  14, 

sin  0°  =^=  0,  cos  0°  =-=  1,  tan  0°  =^=  0,  sec  0°  =-=  1. 
r  r  X  X 

The  definitions  of  cotangent  and   cosecant   given   for  acute 

angles  cannot  be  applied  to  0°  because  y  =  0,  and  therefore  the 

divisions   x/y  and  r/y,  which  occur  in  those  definitions,  are 

impossible. 

Similarly  if  the  angle  of  90°  be   placed  on  the  coordinate 

axes  and  the  construction  of  page  14  be  made,  the  point  P 

will  lie  on  the  y-axis,  and  we  shall  have 

x  =  0,  y  =  'i^' 

The  sine,  cosine,  cotangent,  and  cosecant  of  90°  are  defined 
by  the  same  ratios  as  are  the  corresponding  functions  of  acute 
angles  ;  hence  by  the  definitions 

sin  90°=  ^  =1,  cos  90°  =  -  =  0,  ctn  90°  =  -  =  0,  esc  90°=  -  =  1. 
r  r  y  y 

The  definitions  of  tangent  and  secant  given  for  acute  angles 

cannot  be  applied  to  90°,  because  x  =  0,  and  the  divisions  y/x 

and  7'/x  are  impossible.     We  say  that  0°  has  no  cotangent  or 

cosecant,  and  90°  has  no  tangent  or  secant.* 

23.  Functions  of  30°,  45°,  60°.  In  plane  geometry  it  is 
shown  how  to  construct  a  right  triangle  in  which  one  acute 
angle  is  30°,  or  45°,  or  60°.  From  these  triangles  the  sine, 
cosine,  tangent,  etc.,  of  these  angles  can  be  computed. 


*  It  is  often  said  that  the  tangent  of  90°,  for  example,  is  infinite;  this  ex- 
pression does  not  give  any  value  to  the  tangent  at  90°,  but  merely  describes 
the  fact  that  the  tangent  becomes  and  remains  larger  than  any  number  we 
may  name  as  the  angle  approaches  90°.     Similar  statements  hold  for  the  others. 


30 


PLANE  TRIGONOMETRY 


[HI,  §23 


To  find  the  functions  of  45°,  construct  an  isosceles  right 
triangle  with  the  equal  sides  some  convenient  length  m.  By 
the  Pythagorean  Theorem  compute  the  hypotenuse  =  m^s/% 
Then  by  the  definitions  (10,  11)  §  12, 

m    _    1    _  J    ,_ 


/        \ 

sin  45°= 

/ 

/^                 N 

SsTTl 

and 

/45° 

4X 

cos  45°= 

mV2 

Fig.  26. 

m 


==-A_^i-/o 


m 


V2      V2 


=iV2, 


whence  by  means  of  the  relations  (16,  17,  18,  19),  §  12, 
tan  45°  =  ctn  45°=  1,  and  sec  45°  =  esc  45°=  V2. 
To  find  the  functions  of  30°  and  60°,  construct  an  equilateral 
triangle  of  side  m,  and  divide  it  into  two  right  triangles  by  a  per- 
pendicular from  one  vertex  to  the  opposite  side.  Apply  the 
definitions  (10),  (11),  §  12,  to  obtain  the  values  of  the  functions 
of  30°  and  60°  given  in  the  following  table. 


0^ 

30° 

45° 

60° 

90° 

V2  =  1.414 

V3  =  1.732 

I/V2  =  V2/2 

l/\/3  =  V3/3 

sin 

0 

1/2 

V2/2 

V3/2 

1 

cos 

1 

V3/2 

V2/2 

1/2 

0 

tan 

0 

V3/3 

1 

V3 

These  values  should  be  memorized,  since  the  angles  0°,  30°, 
45°,  60°,  and  90°  occur  frequently.  It  is  easy  to  show  that  all  of 
the  relations  proved  in  §§13,  15,  21,  hold  for  the  values  given 
in  this  table. 

24.  Trigonometric  Equations.  An  equation  that  is  not  an 
identity  (§  21)  is  sometimes  called  a  conditional  equation. 
Thus  the  equation  sin  a  -f  cos  a  =  1  is  not  an  identity  since 
there  are  many  values  of  a  for  which  it  is  not  true ;  there  are 
values   of  a,   however,   which  do   satisfy   the   equation:   for 


Ill,  §25]  TRIGONOMETRIC  EQUATIONS  31 

example,  if  0°  is  substituted  for  a  it  will  be  found  that  the  left- 
hand  members  reduce  to  1  since  sin  0°  =  0  and  cos  0°  =  1. 
This  equation  is  therefore  a  conditional  equation  but  not  an 
identity. 

The  simplest  trigonometric  equations  are  of  the  form 
sin  a  =  1/2,  tan  a  =  1/3,  etc.,  i.e,  equations  in  which  the  angle 
a  is  to  be  determined  from  the  value  of  one  of  the  trigonometric 
ratios.  We  have  already  found  solutions  of  such  equations  in 
Examples  3  and  4,  §  16,  and  Example  1,  §  19.  The  method 
there  employed  of  looking  up  the  value  of  the  angle  in  a 
table  can  always  be  used  for  this  form  of  equation.  A  trigo- 
nometric equation  is  therefore  considered  to  be  practically 
solved  when  it  is  reduced  to  one  of  these  simple  forms.  Eor 
the  present  we  shall  consider  only  positive  solutions  not  greater 
than  90°.  Later  it  will  be  found  that  such  equations  have 
other  solutions.     (See  §§  36  and  68.) 

If  a  trigonometric  equation  contains  more  than  one  of  the 
trigonometric  functions,  all  but  one  can  usually  be  eliminated ; 
the  resulting  equation  may  then  be  solved  algebraically  for 
the  function  which  remains  ;  the  solutions  may  then  be  found 
by  the  methods  explained  above. 

Example  1.  Solve  the  equation  sin2  t  —  cos^  i  =  3  sin  i  —  2.  In  this 
equation  cos2  t  may  be  replaced  by  its  equal  1  —  sin2  t ;  the  equation  then 
becomes  a  quadratic  in  sin  t,  viz. : 

2  sin2  i  ~  3  sin  i  +  1  =0. 

This  equation  is  equivalent  to  the  given  one;  i.e.  every  solution  of  either 
is  a  solution  of  the  other.     The  solutions  may  now  be  found  by  factoring: 

(2  sin  t  -  1)  (sin  «  -  1)  =  0. 
Hence  we  have  either  sin  i  —  1  =  0,  whence  sin  <  =  1,  and  t  =  90°  ;  or 
else,  2  sin  «  -  1  =  0,  whence  sin  ^  =  1/2,  and  t  =  30°.     There  are  no  other 
solutions  which  do  not  exceed  90°. 

25.  Inverse  Functions.  A  notation  is  sometimes  needed  for 
the  angle  whose  sine  (or  any  other  ratio)  is  a  given  number. 
A  notation  quite  frequently  employed  is  sin"^  x  where  x  is  the 
given  number.     In  this  notation  the  equation  sin  a  =  2/7  could 


32  PLANE  TRIGONOMETRY  [III,  §  25 

also  be  written  in  the  form  a  =  sin~^  (2/T).  This  equation  is 
to  be  read,  a  =  the  angle  whose  sine  is  2/7. 

It  should  be  carefully  noted  that  the  (—1)  of  this  notation 
is  not  an  exponent  although  it  is  written  in  the  position 
usually  occupied  by  an  exponent.  Any  other  character 
written  in  the  same  position  would  be  regarded  as  an  ordinary 
exponent;  thus  the  expression  sin^/?  would  be  understood  to 
mean,  the  square  of  the  sine  of  the  angle  /S. 

Many  prefer  the  notation  arcsinx  to  the  one  given  above, 
and  this  notation,  though  not  so  frequently  employed  as  the 
other,  is  nevertheless  used  to  a  considerable  extent.  We  shall 
therefore  throughout  this  book  use  either  notation  in  order  to 
familiarize  the  student  with  both. 

EXERCISES  VIII.  — SIMPLE  TRIGONOMETRIC  EQUATIONS 

1.  Solve  the  following  equations  by  constructing  a  figure  for  each. 

(a)  sin«  =  2/5.  (g)  cosx  =  .63 

(6)  sin  X  =  1/2.  (h)  cos  x  =  V3/2. 

(c)  sin  ic  =  .8  (i)  sin  x  =  0. 

(d)  sin  aj  =  .866  (j)  cos  x  =  0. 

(e)  sin  x  =  AS  (Jc)  sin  x  =  1. 
(/)  cosx  =  1/2.  (I)  cosx  =  1. 

2.  Prove  that  there  is  always  an  acute  angle  solution  of  the  equation 
sin  x=c,  if  c  is  any  number  between  0  and  1. 

3.  Prove  that  there  is  always  an  acute  angle  solution  of  the  equation 
tan  X  =  c,  if  c  is  any  positive  number  whatever. 

4.  Find  sin-i  (2/5)  graphically. 
[Hint.     Compare  Ex.  1(a).] 

5.  Express  the  answer  to  each  of  the  exercises  1(a)  to  \(l)  by 
means  of  the  notation  sin-i  or  cos-i  (or  arcsin,  arccos,  etc.). 

6.  Find  sin-i(2/3),  and  also  tan-i  (1/2)  graphically. 

7.  Find  arcsin  (.66667),  and  also  tan-i  (.60000)  by  the  Tables. 
Solve  each  of  the  following  equations  for  x. 

8.  2  sin2  X  +  sin  X  =  1. 

[Hint.  Solve  this  quadratic  for  sin  x.  There  are,  of  course,  no  solu- 
tions corresponding  to  values  of  sin  x  greater  than  1.] 

9.  (a)  2  sin2  x  —  5  sin  x  +  2  =  0.  (6)  4  cos2  ^  -f  8  cos  ^  =  5. 


Ill,  §25] 


TRIGONOMETRIC  EQUATIONS 


33 


10.  (a)tanaj=l.  (d)  tan  x  =— 2.6 
(6)  tanx  =—  1/2.                                   (e)  tan  x  =  5.3 
(c)  tan  X  =  2.                                           (/)  tan  x  =  0. 

11.  (a)taii2x  =  3.  (6)tan2^  =  6J.  (c)  tan2  (?  =  6  -  4  V27 

12.  2  sin2  X  —  cos  x  =  1.  13.    cos2  x  =  sin2  x. 
14.   5  sin  X  +  2  cos2  x  =  5.                           15.    sec2  x  +  tan  x  =  3. 

16.  If  a  and  b  are  the  sides  of  a  right  triangle,  c  the  hypotenuse,  and  A 
the  angle  opposite  a,  show  that  the  area  of  the  triangle  is  equal  to  either 
of  the  expressions 

ac  cos  A     he  sin  A 


Fig.  27. 


17.  Two  straight  pieces  of  railroad  track  MA  and  NB  are  to  be  con- 
nected by  a  circular  track  AKB  with  a  radius  of  500  ft.  and  center,  0, 
tangent  to  MA  and  NB.  The 
straight  portions  of  the  track  pro- 
duced intersect  at  a  point  V  at  an 
angle  of  100°. 

(a)  How  far  back  from  F should 
the  track  begin  to  turn  ? 

(6)  How  far  from  V  along  the 
, bisector  OF  of  the  angle  AVB  is 
the  center  0  ? 

(c)  Find  the  shortest  distance 
from  V  to  the  curved  portion. 

18.  If,  in  a  figure  similar  to  that  of  Ex.  17,  Z.  AVO  is  any  angle,  and 
Z  VOA  is  denoted  by  a,  and  OA  =  r,  show  that 

(a)  J.F=  r  tana; 
(6)  -fiTF  =  r  exsec  a  ; 
(c)   ^-B  =  2  r  sin  a. 

19.  The  side  b  of  the  triangle  in  Ex.  16  is  extended  beyond  ^  to  a 
point  D,  making  AD  =  c,  so  that  ^-BD  is  isosceles.     Show  that 

(a)  ZADB]=A/2; 
(6)  ^i)  =  2^"ccos  (^/2). 
(c)   From  the  right  triangles  DOB  and  ACB, 
show  that 

c  sin  J.  =  a  =  2  c  cos  (A/2)  sin  (^/2)  ; 
hence 

sin  -4  =  2  sin  (A/2)  cos  (^/2)  ; 

(d)  Likewise,  show  that  c  cos  A=b  =  2  c  cos2  (A/2)  —  c  ; 
hence        cos  A  =  2  cos2  (A/2)  -1  =  cos2  (A/2)  ~  sin2  (A/2). 


34 


PLANE  TRIGONOMETRY 


nil,  §27 


Fig.  29. 


26.  Projections.  The  projection  of  a  line  segment  AB  upon 
a  line  I  is  defined  to  be  the  portion  MN  of  the  line  I  between 
perpendiculars  drawn  to  it  from  A  and  B,  respectively.     The 

length  of  this  projection  is  easily  found 
if  the  length  of  AB  and  the  angle  a 
which  the  line  AB  makes  with  I  are 
known.  Eor,  draw  a  parallel  to  I  through 
A,  meeting  BN  at  C.  Then  AOB  is  a 
right  triangle  and  the  angle  at  ^  is  a ; 
hence  by  (14),  §  12, 
MN  =  AB  cos  a 

or,  the  projection  of  a  segment  upon  a  given  line  is  equal  to  the 

product  of  the  length  of  the  segment  and  the  cosine  of  the  angle  the 

segment  makes  with  the  given  line. 
The  projections  of  a  segment  upon 

the  coordinate  axes  are  frequently 

used.      If  the   segment    makes  an 

angle  a  with  the  horizontal,  the  pro- 
jections on  the  X  and  y  axes  are, 

respectively, 

(5)  Tto]^AB  =  AB  cos  a, 
Projj,  AB  =  AB  sin  a, 

where  Tto^^AB  and  'Pro] ^AB  denote  the  projections  of  AB  on 

the  a>axis  and  the  y-axis,  respectively. 

27.  Applications  of  Projections.  In  mechanics  and  related 
subjects,  forces  and  velocities  are  represented  graphically  by 
line  segments.  A  force,  say  of  10  lb.,  is  represented  by  a  seg- 
ment 10  units  in  length  in  the  direction  of  the  force.  A  veloc- 
ity of  20  ft.  per  sec.  is  represented  by  a  segment  20  units  in 
length  in  the  direction  of  motion. 

The  projection  upon  a  given  line  ?,  of  a  segment  represent- 
ing a  force,  represents  the  effective  force  in  the  direction  I ;  this 
is  called  the  component  of  the  given  force  in  the  direction  L 


Fia.  30. 


Ill,  §27] 


PROJECTIONS 


35 


Example  1.  A  weight  of  50  lb.  is  placed  upon  a  smooth  plane  in- 
clined at  an  angle  of  27°  with  the  horizontal.  What  force  acting  directly 
up  the  incline  will  be  required  to  keep  the  weight 
at  rest  ? 

Draw  to  some  convenient  scale  a  segment  60 
units  in  length  directly  downward  to  represent 
the  force  exerted  by  the  weight.  Projectjthis  seg- 
ment upon^a  line  inclined  at  an  angle  of  27°  with 
the  horizontal.  The  length  of  this  projection  WQ, 
Fig.  31,  is  50  cos  63°  =  22.7  nearly.  This  repre- 
sents the  component  of  the  force  down  the  plane. 


Fig.  31. 

Therefore,  a  force  of  22.7  lb.  acting  up  the  plane  will  be  required. 


Example  2.     A  ladder  30  ft.  long,  when  lying  horizontal  supported  at 
its  ends,  will  carry  a  safe  load  of  150[lb.  on  its  middle  round.     Is  it  safe 
for  a  man  weighing  190  lb.  to  mount  it  when  it 
is  so  placed  as  to  reach  a  window  18  ft.  above 
the  ground  ? 

We  have  to  find  the  component,  perpendicu- 
lar to  the  ladder,  of  the  man's  weight  when  he 
stands  on  the  middle  round.  Let  TTP,  drawn 
vertically  downward  from  the  middle  point 
of  AB,  Fig.  32,  represent  190  (which  need 
not  be  on  the  same  scale  as  ^J5  which  repre- 
sents 30) .  Then  the  component  perpendicular 
to  AB  is 


Fig 


Now  by  (11)  §  12, 
whence 


WQ  =  190  cos  PWQ  =  190  cos  GAB, 
cos  CAB  =  AC/AB  =  4/6, 


WQ  = 


190  X  4 . 
6 


:162, 


which  is  greater  than  the  safe  load. 

Example  3.  A  traveling  crane  moves  with  uniform  speed  down  a 
shop  297  ft.  long  and  60  ft.  wide  in  1  min.  41  sec.  It  carries  a  load  from 
one  corner  along  the  diagonal  to  the  opposite  corner.  Find  the  speed  of 
the  crane  and  of  the  car  which 
runs  on  it. 

Let  AP  =  the  speed  of  the  load 
along  the  diagonal  which  by  the 
data  of  the  problem  =  3  ft.  per 
sec.  (AP  need  not  of  course  be  on 
the  same  scale  as  ^5  and  AD). 
3  cos  PAQ  =  2.94+  and  QP 


Fig.  33. 


Then  AQ  =  the  speed  of  the  crane  = 
the  speed  of  the  car  =  3  sin  PAQ  =  .69+ 


36  PLANE  TRIGONOMETRY  pil,  §  27 

EXERCISES  IX.— PROJECTIONS 

1.  Find  the  horizontal  and  vertical  projections  of  the  segments  : 
(a)  length  42,  making  an  angle  of  37°  with  the  horizontal. 

(6)  length  5.5,  making  an  angle  of  50°  with  the  vertical. 

Ans.  (a)33.54,  26.28  ;  (6)  3.54,  4.21 

2.  A  straight  railroad  crosses  two  north  and  south  roadways  a  mile 
apart.  The  length  of  track  between  the  roadways  is  1 J  mi.  A  train 
travels  this  distance  in  2  min.  Find  the  components  of  the  velocity  of 
the  train  parallel  to  the  roadways  and  perpendicular  to  them.  Find  the 
angle  between  the  track  and  either  roadway.  Ans.  |,  J,  53°  7.8' 

3.  The  eastward  velocity  of  a  certain  train  is  24  mi.  per  hour.  The 
northward  velocity  is  32  mi.  per  hour.  Find  its  actual  velocity  along  the 
track  and  the  angle  the  track  makes  with  the  east  and  west  direction. 

Ans.  40,  53°  7.8' 

4.  A  car  is  drawn  by  means  of  a  cable.  If  a  force  of  5000  lb.  exerted 
along  the  track  is  required  to  pull  the  car,  what  force  will  be  required 
when  the  cable  makes  an  angle  of  15°  with  the  track  ?  Ans.  5176.4 

5.  Find  the  horizontal  and  vertical  components  of  a  force  of  30  lb. 
making  an  angle  of  40°  with  the  horizontal.  Ans.  22.98,  19.28 

6.  Find  the  horizontal  and  vertical  projections  of  the  segment  which 
joins  the  points  (8,  —  3)  and  (—2,  7).  Ans.  10,  10. 

7.  The  stringers  for  a  stairway  are  20  ft.  7.8  in.  long.  The  steps  are 
to  have  7  in.  risers  and  12  in.  treads  (which  includes  1  in.  overhang) . 
Determine  the  number  of  steps,  using  the  horizontal  and  vertical  projec- 
tions of  the  stringer  to  check  the  result.  Ans.  19. 

8.  Five  forces  act  on  the  point  A:  (—4,  0)  viz.:  AB,  AC,  AD,  AE, 
AF,  and  the  points  A,  B,  (7,  D,  E,  F  are  the  vertices  of  a  regular  hexa- 
gon, center  at  the  origin.     Show  that  the  vertical  com- 
ponents balance,  and  find  the  sum  of  the  horizontal 
components.  Ans.  24. 

9.  Determine  the  width  and  height  of  a  crate  for 
the  chair  shovm  in  Fig.  34.  Ans.  35+,  48f+. 

10.   In  surveying,   the  projection  of  a  line  on  a 

north  and  south  line  is  called  the  latitude  of  the  line 

and  the  projection  on  an  east  and  west  line  is  called 

the  departure  of  the  line.    Find  the  latitude  and  de- 

Fig   34 
parture  of  the  following  lines: 

(a)  length  41  rods,  bearing  N  26°  15'  E.  Ans.  36.772,  18.134 

(6)  length  487  feet,  bearing  E  32°  30'  S.  Ans.  259.66,  410.73 

(c)  length  17.32  rods,  bearing  N  40°  45'  W.  Ans.  13.053,  11.247 


CHAPTER   IV 
LOGARITHMIC    SOLUTIONS    OF    RIGHT   TRIANGLES 

28.  The  Use  of  Logarithms.  Logarithms  may  be  used  to 
shorten  computations  involving  multiplications ,  divisions,  rais- 
ing to  powers  or  extracting  roots,  but  not  involving  additions  or 
subtractions.  In  much  of  the  numerical  work  which  follows, 
the  use  of  logarithms  is  very  advantageous  in  saving  time  and 
labor,  but  the  student  should  bear  in  mind  that  logarithms  are 
not  necessary.  They  are  merely  convenient,  and  they  belong 
no  more  to  trigonometry  than  to  arithmetic.  One  of  the  ques- 
tions which  a  computer  has  to  decide  is  whether  or  not  it  will 
be  advantageous  to  use  logarithms  in  a  given  problem. 

At  the  end  of  this  book  will  be  found  a  table  of  the 
logarithms  of  numbers  (Tables,  p.  1),  and  a  table  of  the 
logarithms  of  the  trigonometric  functions  (Tables,  p.  45),  with 
explanations  of  their  use  (pp.  v-xvii)."*  In  case  a  review  of 
the  principles  of  logarithms  is  desired,  this  explanation  should 
be  studied  before  proceeding  with  the  rest  of  this  chapter. 

The  notation  log  tan  62°  51'  means  the  logarithm  of  the 
tangent  of  62°  51' ;  the  tangent  of  62°  51'  is  a  number,  1.9500, 
and  the  logarithm  of  this  number  is  0.29003,  as  may  be  seen 
by  looking  up  log  1.9500  in  Table  I.  This  last  result  is  found 
in  Table  III,  p.  73,  which  enables  us  to  avoid  the  labor  of 
looking  in  Tables  II  and  I,  in  succession. 

A  formula  which  has  been  arranged  so  as  to  involve  only 
products  and  quotients  of  powers  and  roots  of  quantities 
either  known  or  easily  computed  from  the  known  quantities, 

*  In  the  edition  of  this  book  with  brief  tables,  only  four-place  tables  are 
given.  Those  using  that  edition  should  refer  to  The  Macmillan  Tables, 
to  which  all  page  references  made  here  apply. 

37 


38 


PLANE  TRIGONOMETRY 


[IV,  §29 


is  said  to  be  adapted  to  loganthmic  computation. 


Thus  the  formula  h  =  va2  +  62,  which  gives  the  hypotenuse  /i  of  a 
right  triangle  in  terms  of  the  sides  a  and  6,  is  not  adapted  to  logarithmic 
computation.     On  the  other  hand,  the  formula 


6=V/i2-a2=>/(/i  +  a)  Qh  —  a) 

which  gives  one  side  in  terms  of  the  hy- 
potenuse and  the  other  side,  is  adapted  to 
logarithmic  computation  because  (/i  -f  a) 
and  (/i—  a)  are  easily  obtained  from  h  and 
a.  Thus,  if  the  hypotenuse  is  17.34  and 
one  side  is  12.27,  the  other  side  is 


x  =  V(5.07)  (29.61) 
log  5.07  =0.70501 
log  29.61  =  1.47144 

log  X2  : 


Tables,  p.  10 
Tables,  p.  5 


:  2.17645 
logx  =  1.08822 

X  =  12.252  Tables,  p.  2 

The  formulas  (10  to  19),  §§  12,  13,  are  all  adapted  to  loga- 
rithmic computation. 

Example  1.     Find  a  =  29.45  sin  46°  23 

log  29.45  =  1.46909  Tables,  p.  5 

log  sin  46°  23'  =  9.85972  -  10  Tables,  p.  89 

log  a  =  1.32881 

a  =  21.321  Tables,  p.  4 

675.4 


Example  2.     Find  a  from  tan  a  -. 

log  675.4  =  2.82956 

log  423.7  =  2.62706 

log  tan  a  =  0.26250 

a  =  57°  53'.9 

42.98 


423.7 


Example  3.     Find  h  = 


cos  15°  20' 
log  42.98  =  11.63327 -10 
log  cos  15°  20'  =    9.98426  -  10 
logh=    1.64901 
h  =  44.567 


Tables,  p.  13 
Tables,  p.  8 

Tables,  p.  78 


Tables,  p.  8 
Tables,  p.  61 


Tables,  p.  8 

29.  Products  with  Negative  Factors.  To  find  by  use  of 
logarithms  the  product  of  several  factors  some  of  which  are 
negative,  the  product  of  the  same  factors,  all  taken  positively, 
is  first  obtained,  and  the  sign  is  then  determined  in  the  usual 


IV,  §  29]    RIGHT  TRIANGLES  BY  LOGARITHMS 


39 


manner  by  counting  the  number  of  factors  with  negative  sign. 

Example  1.     Find  x  =  {-  115)  (23.41)  (-  .6422)  (-  .1123) 
Noticing  first  that  there  are  an  odd  number  of  negative  factors,  we  may 

^^^*^  -x  =  (115)  (23.41)  (.6422)  (.1123); 

and  we  may  compute  —  x  as  follows. 

log  115  =  2.06070 
log  23.41  =  1.36940 
log  .6422  =  9.80767  -  10 
log  .1123  =  9.05038  -  10 
log  (-X)  =2.28815 

—  X  =  194.15  whence  x  =  —  194.15 

The  use  of  logarithms  in  numerical  calculation  is  further  illustrated  in 
the  following  examples. 


Example  2.     Find  x 


-f 


740050 

2  log  87  =    3.87904 

i  log  3241  =    1.75534 

5.63438 

5.86926 


log  740050 

log  X3 


whence 


29.76512-30 
logx=    9.92171-10 
x=    0.83504 


o      -I.-   ^  /5.62(4.8)i-' 

Example  3.     Findx^x/    /  aWs 


whence 


\-  (.e 

log  5.62=    0.74974 
1.5  log  4.8=    1.02186 

11.77160-10 
2.3  log  0.684  =    9.62064-10 
^   logx2=    2.15096 
logx=    1.07548 
x  =  11.898 


Tables,  p.  17 
Tables,  p.  6 

Tables,  p.  14 


Tables,  p.  16 


Tables,  p.  10 
Tables,  p.  9 

Tables,  p.  13 


Tables,  p.  2 


EXERCISES  X.  — LOGARITHMS.    RIGHT  TRIANGLES 

1.    Make  the  following  computations  by  logarithms 

(a)   .001467  X  96.8  x  47.37  Ans.  6.7268 

(6)   .0631  X  7.208  x  .51272  Am.  0.23317 

(c)  2v^5/3^  Ans.  0.1364 

(d)  \/-  0.00951  Ans.  -0.5142 

(e)  15.008  X  (-  0.0843)7(0.06376  x  4.248)  Ans.  -  4.671 
(/)  y/EM^x  \/6l72/v/298:54  Ans.  3.076 


40  PLANE  TRIGONOMETRY  [IV,  §  29 

(9)  (18.9503)11  (-O.l)i^  Ans.  1.134 

(h)  (-  0.1412)2/^-0.00476  Ans.  -  0.11858 

(i)    1/(72.32)J  Ans.  0.05761 

(j)  V(0-00812)*  (471.2)vV(522.3)8  (0.01242)*  Ans.  0.8929 

2.  The  following  formula  d  =  0.479-v/— r^  is  used  to  determine  the 

diameter  d,  of  water  pipe  in  terms  of  the  coefficient  of  friction  c,  the 
length  I,  the  flow/,  and  the  head  h.  Compute  d  when  c  =  0.02,  I  =  500, 
/zi:5,  /t=:10.  ^ns.  0.91136 

3.  A  wire  0.1066  cm.  in  diameter  and  27.1  cm.  long  is  stretched  0.133 
cm.  by  a  weight  of  454  grams.     Find  the  modulus  of  elasticity  by  the 

formula    e  =  — ,   in  which  I  =  length,  a  —  area  of  cross  section,  and  s  = 

the  elongation  produced  by  a  weight  w.  Ans.  1.0365  x  10^. 

4.  The  flow  of  water  over  a  weir  is  given  by  the  formula 

/=  ^V2^6 

Find/ when  k  =  4.736,  g  =  32.2,  h  =  1.2  Ans.  399.32 

5.  A  steel  bar  98.75  cm.  long  between  supports  0.96  cm.  wide  and 
0.74  cm.  deep  is  deflected  1.48  cm.  by  a  weight  of  5000  grams  at  the  middle. 

Find  the  modulus  of  elasticity  by  the  formula  e  =  .  ,  ,„,,   in  which  I  = 

4  od^fi 

length,  b  =  breadth,  d  =  depth,  and  h  =  the  deflection  due  to  the  weight  w. 

Ans.  2.0908  x  109. 

6.  The  pressure  p  and  the  volume  v  of  a  gas  at  constant  tempera- 
ture are  connected  by  the  relation  pv^  =  k.  Find  p  when  v  =  36.36, 
a  =  1.41,  k  =  12600.  Ans.  79.414 

7.  The    period    of   a  conical   pendulum   is  given   by  the    formula 

T  =  2t\^^^^^^.     Find  T  when  m  =  0.347,  I  =  96.8,  a  =  9°  20^  w  =  340. 

Ans.  1.9618 

8.  The  volume  (gal.)  of  a  conical  tank  of  height  h  (in.)  and  vertical 
angle  2  ais  v  =  irh^  tan^  06/693.  Find  the  capacity  of  such  a  tank  whose 
angle  at  the  vertex  is  42°  30'  and  whose  height  is  12  ft.  5  in. 

Ans.  2267.8 

9.  If  a  ball  of  radius  r  is  rolled  inside  a  spherical  surface  of  radius  i?, 

the  time  of  oscillation  is  given  by  the  formula  T  =  2ir'^ — ^ -.     Find 

the  radius  of  a  concave  mirror  in  which  a  |  in.  steel  ball  makes  an  oscilla- 
tion in  1.4  sec.     Take  g  =  384.  Ans.  13.805 


rv,  §  29]    RIGHT  TRIANGLES  BY  LOGARITHMS  41 

10.  Solve  by  means  of  logarithms  the  following  right  triangles,  where 
h  denotes  the  hypotenuse,  other  small  letters  the  sides,  and  the  corre- 
sponding capital  letters  the  angles  opposite  those  sides. 

(a)  J.  =  63°  ;  h  =  28.54  Ans,  25.429,  12.957 

(6)  P  =  65°  25'.2  ;  p  =  69.25  Arts.  31.676,  76.152 

(c)  A  =  28°  25' ;  h  =  29.36  Ans.  25.822,  13.972 

Id)  Cr=  28°  40'.4  ;  v  =  20.71  Ans,  11.326,  23.605 

(e)    a  =  735.1  ;  h  =  846.2  Ans,  60°  18^6,  419.14 

(/)    r  =  9.328  ;  s  =  6.302  Ans.  55°  57^4,  11.257 

(g)    a  =  59.68  ;  h  =  69.27  Ans,  59°  29^4,  35.17 

(/i)  G  =  36°  21'  ;  /i,  =  41.376  Ans,  33.325,  24.524 

11.  Solve  the  following  right  triangles  having  given 

(a)  hypotenuse  =  431.8,  side  =  127.3  Ans,  17°  8'. 7,  412.61 

lb)  angle  =  43°  48^  side  adj.  =  67.92  Ans,  94.104,  65.133 

(c)  angle  =  55°  11',  side  opp.  =  68.34  Ans.  83.242,  47.527 

(d)  hyp.  =  61.14,  side  =  48.56  Ans,  37°  25',  37.149 

(e)  angle  =  49°  13',  side  adj.  =  72.3  Ans.  110.68,  83.810 
(/)  sides  =  126  and  198.  Ans.  234.72,  32°  28J'. 
(g)  angle  =  57°  46',  side  opp.  ^  0.688  Ans.  0.4338,  0.8134 
Ih)  angle  =  32°  15'.4,  side  opp.  =  547.25  Ans,  867.12,  1025.4 

12.  A  tree  stands  on  the  opposite  side  of  a  small  lake  from  an  observer. 
At  the  edge  of  the  lake  the  angle  of  elevation  of  the  top  of  the  tree  is 
found  to  be  30°  58'.  The  observer  then  measures  100  ft.  directly  away 
from  the  tree  and  finds  the  angle  of  elevation  to  be  18°  26'.  Find  the 
height  of  the  tree  and  the  width  of  the  lake.  Ans,  74.973,  124.94 

13.  From  a  point  250  ft.  from  the  base  of  a  tower  on  a  level  with  the 
base  the  angle  of  elevation  of  the  top  is  62°  32'.     Find  the  height. 

Ans,  480.93 

14.  To  determine  the  height  of  a  tower,  its  shadow  is  measured  and 
found  to  be  97.4  ft.  long.  A  ten-foot  pole  is  then  held  in  vertical  position 
and  its  shadow  is  found  to  be  5.5  ft.  Find  the  height  of  the  tower  and 
the  angle  of  elevation  of  the  sun.  Ans,  177.09,  61°  11'.4 

15.  Find  the  length  of  a  ladder  required  to  reach  the  top  of  a  building 
50  ft.  high  from  a  point  20  ft.  in  front  of  the  building.  What  angle  would 
the  ladder  in  this  position  make  with  the  ground  ?    Ans.  53.85,  68°  12'. 

16.  The  width  of  the  gable  of  a  house  is  34  ft.  ;  the  height  of  the  house 
above  the  eaves  is  15  ft.  Find  the  length  of  the  rafters  and  the  angle  of 
inchnation  of  the  roof.  Ans.  22.67,  41°  25'.4 

17.  Assuming  the  radius  of  the  earth  to  be  3956  mi.  find  the  distance 
to  the  remotest  point  on  the  surface  visible  from  the  top  of  a  mountain 
2J  mi.  high.  Ans.  140.67  mi. 


CHAPTER  V 

SOLUTION   OF    OBLIQUE    TIOANGLES  BY  MEANS   OF 
RIGHT  TRLA.NGLES 

30.  Decomposition  of  Oblique  Triangles  into  Right  Tri- 
angles. A  general  method  for  solving  oblique  triangles  in  all 
cases  consists  in  dividing  the  triangle  into  two  right  triangles 
by  a  perpendicular  from  a  vertex  to  the  opposite  side ;  these 
right  triangles  are  then  solved  by  the  methods  of  the  previous 
chapter.  In  all  except  the  three  side  case  the  perpendicular 
can  be  drawn  so  that  one  of  the  resulting  right  triangles  con- 
tains two  of  the  given  parts.  It  may  sometimes  happen  that 
the  perpendicular  will  fall  outside  the  given  triangle. 

31.  Case  I :  Given  Two  Angles  and  a  Side.  It  is  im- 
material which  side  is  given,  since  the  third  angle  can  be  found 
from  the  fact  that  the  sum  of  the  three  angles  is  180°.  Drop 
the  perpendicular  from  either  extremity  of  the  given  side. 

Example  1.     An  oblique  triangle  has  one  angle  equal  to  43°,  another 
equal  to  67°,  and  the  side  opposite  the  unl«iown  angle  equal  to  61.     De- 
^     termine  the  remaining  parts. 

It  is  immediately  seen  that  the  third  angle  is 
180°-(43°  +  67°)  =  70^.  To  solve  this  triangle  draw 
the  figure  approximately  to  scale  and  drop  the  perpen- 
^^  dicular  CD=p  from  one  extremity  C  of  the  known 
side  to  AB,  the  side  opposite  C.  Denote  the  unknown 
side  CB  by  a.  In  the  right  triangle  J.  CD,  the  hypot- 
^A  enuse  and  one  angle  are  known  ;  hence  by  (13),  §  12, 
p  =  51  sin  67°  =  46.95 
An  angle  and  the  side  opposite,  in  the  right  triangle  BCD,  are  now 
known;  hence  by  (15),  §  12, 

a  =  p/sin  70°  =  46.95/.9397  =  49.96 
The  side  AB  may  be  found  in  the  same  manner.     Check  as  in  §  5,  p.  4. 

42 


V,§32]       SOLUTION  OF  OBLIQUE  TRIANGLES 


43 


If  in  the  equation  a  =  p/sin  70°  we  substitute  the  value 
p  =  51  sin  67°  previously  found,  we  obtain  for  a  the  equation 

51  sin  67° 
"*-     sin  70° 

This  formula  is  adapted  to  logarithmic  computation.  Apply- 
ing the  principles  of  logarithms  we  obtain 

log  a  =  log  51  +  log  sin  67°  —  log  sin  70°. 
Eemembering  that  subtracting  a  logarithm   is   equivalent   to 
adding  the  co-logarithm  of  the  same  number,  we  may  arrange 
the  numerical  work  as  follows  : 

log  51  =  1.70757 
log  sin  67'^  =  9.96403 -10 
colog  sin  70°  =  0.02701 
log  a  =  1.69861 
a  =  49.959 

In  this  solution,  p  was  eliminated.  Even  if  the  equations 
are  used  without  eliminating  p^  the  actual  value  of  p  need  not 
be  found,  since  only  log  p  is  needed  to  complete  the  solution. 

32.  Case  II :  Given  Two  Sides  and  the  Included  Angle. 

The  triangle  can  be  divided  into  two  right  triangles,  one  of 
which  contains  two  known  parts,  by  a  perpendicular  from 
eithei'  extremity  of  the  unknown  side  to  the  side  opposite. 

Example  1.  Two  sides  of  a  triangle  are  26.5  and  32.8  ;  the  included 
angle  is  52°  18'.     Find  the  remaining  parts. 

In  the  figure  let  J.5  =  32.8,  ^.0  =  26.5, 
and  the  angle  at  J.  =  52°  18'.  Drop  a  per- 
pendicular p  from  B  to  the  opposite  side. 
Denote  the  unknown  side  by  a  and  the  seg- 
ments of  ACbj  X  and  y  as  in  Fig.  37  ;  then 
p,  x,  ?/,  and  tan  G  can  be  computed  in  the 
following  order : 

p  =  32.8  sin  52°  18'  =  32.8  x  .79122  =  25.952 

X  =  32.8  cos  52°  18'  =  32.8  x  .61153  =  20.058 

y  =  26.5  -x  =  26.5  --  20.058  =  6.442 

tan  C  =p  H-  2/  =  25.952  x  6.442  =  4.0286 


44  PLANE  TRIGONOMETRY  [V,§33 

Hence  from  the  tables, 

(7=76°  3'.6 

a  =  y-^cosG  =  6.442  --  .24101  =  26.73 

These  formulas  are  not  well  adapted  to  logarithmic  compu- 
tation. The  values  of  p  and  x  may  be  computed  separately 
by  logarithms,  after  which  y  and  tan  C  may  be  found. 

We  use  the  formulas  p  =  c  sin  A,  x  =  c  cos  A,  y  =  b  ^  Xy 
tan  C  =  p  -T-  y.  The  work  can  be  conveniently  arranged  in 
two  columns,  as  follows  : 

log  32.8  =  1.61687  log  32.8  =  1.61687 

log  sin  A  =  9.89830  log  cos  A  =  9.78642 

logp  =  1.41417  log  X  =  1.30229 

log  y  =  0.80902  x  =  20.068 

log  tan  C  =  0.60515  y  =  h^x=  6.442 

C=76°3'.6  \ogy=  0.80902 

a  =  y-^  cos  G  log  cos  C  =  9.38190 

a  =  26.738  log  a  =  1.42712 

33.  Case  III :  Given  the  Three  Sides.  In  this  case  it  is  not 
possible  to  divide  the  triangle  into  two  right  triangles  in  such 
a  way  that  one  of  them  contains  two  of  the  given  parts ;  how- 
ever, if  a  perpendicular  is  dropped  to  the  longest  side  from 
the  vertex  of  the  angle  opposite,  the  segments  into  which  this 
side  is  divided   by   the   perpendicular  are   easily  computed. 

g  Example  1.     The  sides  of  a  triangle  are 

>         ^/TV        •  a  =  36.4,  6  =  50.8,  and  c  =72.6    Determine 

<^^  p,   ^^y  ^^^  angles. 
^y^             I         ^^y  Draw  a  figure  and  drop  a  perpendicular 

A-^^- 72^5  ^^  from  B  upon  AG,     Denote  the  segments  of 

Fig.  38.  '  the  base  by  x  and  y  as  in  Fig.  38  ;  then 

p2  =  50.8^  -  x2  =  36^^  -  y2  ; 

hence  ajz  _  2/2  =  608^  -  36^^^  =  1266.68 ; 

that  is,  {x  -y)  (x  +  y)  =  1266.68 

Since  x  +  y  =  b  =  72.6, 

we  have  x-y  =  1266.68  -^  72.6  =  17.32  ; 

whence,  adding,  x  =  44.91, 

and,  subtracting,  2/  =  27.69 


V,§34]        SOLUTION  OF  OBLIQUE  TRIANGLES 


45 


Since  we  now  know  x  and  ?/,  the  angles  A  and  C  are  easily  found. 
The  student  may  complete  the  solution  by  using  the  formulas 
cos  ^  =  aj  -f-  50.8  cos  G  =  y  -^  36.4 

Logarithms  may  be  used  as  in  the  previous  case  to  compute 
the  separate  products  and  quotients.  The  following  is  a  con- 
venient arrangement : 


x2  ^y2  =  50.8'  -  36.r  =  c2  -  a2. 
Factoring  both  sides  gives 

(x  4-  y)  (X  —  y)  =  b(x-y)  =  (c-{-  a)  (c  - 

X  —  ?/  =  (c  4-  a)  (c  —  a)  -f-  6 


a) 


c  =  50.8 
a  =  36.4 
c  -f  a  =  87.2 
c  —  a  =  14.4 
x  +  y=zb  =  72.5 
X- 2/ =  17.32 
X  =  44.91 
2/  =  27.59 
cos  A  =  x  -^  c 
log  X  =  1.65234 
log  c  =  1.70586 
log  cos  ^  =  9.94648 
A  =  27°  51^9 


5  = 


log  (c  -I-  a)  =  1.94062 

log  (c-a)  =1.15836 

colog  b  =  8.13966 

log  (x-y)  =  1.23854 


cos  C  =  y  -^  a 
log  y  =  1.44075 
log  a  =  1.56110 
log  cos  C  =  9.87965 
C  =  40°42'.9 
111°  25'.2 


34.  Case  IV :  Given  Two  Sides  and  the  Angle  Opposite 
One  of  Them.  The  triangle  is  solved  by  dropping  the  perpen- 
dicular from  the  vertex  of  the  angle  included  hy  the  given  sides. 

Example  1.  One  angle  of  a  triangle  is  37""  20'  ;  one  side  adjacent  is 
25.8  and  the  side  opposite  is  20.8.     Solve  the  triangle. 

First  construct  the  given  angle  A 
and  on  one  side  of  A  lay  off  ^5  =  25.8 
With  B  as  center  and  radius  =  20.8 
describe  an  arc  of  a  circle  meeting 
the  opposite  side  in  two  points  C  and 
C^  Either  of  the  triangles  ABC, 
ABC  satisfies  the  given  conditions; 
the  case  is  on  this  account  called  the 
ambiguous  case.  Fio.  39. 


46  PLANE  TRIGONOMETRY  [V,§34 

The  student  should  note  that  the  triangle  5  CC  is  isosceles  and  that 
the  interior  angle  of  ABC  at  C  is  equal  to  the  exterior  angle  of  ABC 
at  C ;  hence  the  interior  angles  C  and  C  are  supplements  of  each  other. 
To  solve  ABC  draw  the  perpendicular  BD  =p  from  B;  then  determine 
p  from  the  right  triangle  ABB. 

p  =  25.8  sin  37°  20'  =  15.6464 
Next  determine  C  from  the  right  triangle  BD  G; 

.     ^     p      15.6464       ^^„„^ 
«^^^  =  a  =  -20:8-=-^^''^' 

hence  C  is  the  acute  angle  whose  sine  is  .75223  ;  i.e.  (7=48°  47'. 
The  student  can  complete  the  solution  as  follows: 
AC  =  AD  +  DC; 
B  =  180°  -(A+  C). 
Also  for  triangle  ABC ^ 

a  =  180°  -  C ; 
•5' =  180°-  (^+  C); 
AC  =  AD -CD. 

For  the  logarithmic  solution  we  use  the  formula 

.     n     P     <^  sin  A 

sm  C  =  -= 

a  a 

Then  the  work  may  be  arranged  as  follows : 

logc  =  1.41162 

log  sin  A  =  9.78280 

colog  a  =  8.68194 

log  sin  (7=9.87636 

C  =  48°  47M  C  =  131°  12'.9 

5  =  93°52'.9  jB'  =  11°27M 

6  =  a  sin  B/sin  A  b'  =  a  sin  B'  /  sin  A 

log  a  =    1.31806  log  a  =  1.31806 

log  sin  B  =    9.99900  log  sin  B'  =  9.29785 

colog  sin  A  =    0.21720  colog  sin  A  =  0.21720 

log  6=    1.53426  log  6' =  0.83311 

h  =  34.218  b'  =  6.8094 

If,  in  a  given  problem,  the  side  opposite  the  given  angle  is 
less  than  the  perpendicular  let  fall  upon' the  unknown  side, 
there  is  no  solution,  and  if  it  is  greater  than  the  other  given 
side  there  is  one  solution  only.  The  construction  indicated  in 
Ex.  1  will  in  all  cases  show  the  number  of  solutions. 


V,§34]        SOLUTION  OF  OBLIQUE  TRIANGLES  47 

EXERCISES   XL  — SOLUTION   OF  TRIANGLES 

Find  the  remaining  parts  of  the  following  triangles  by  suitably  divid- 
ing each  into  two  right  triangles.  Capital  letters  represent  angles  ;  small 
letters  the  sides  opposite  them. 

1.  (a)  A  =  17°  17',  B  =  Sr  37^  c  =  174  ;  Ans.  63.186,  129.81 
(6)  A  =  24°  14^  a  =  43"  13',  c  =  240  ;  Ans,  143.86,  323.69 
(c)   L  =  28°,         M  =  51°,  I  =  6.3  Ans.  10.429,  13.173 

2.  (a)    a  =  41,    6  =  51,    C  =  62°  ;    Am.  48°  44'.7,  69°  15' .3,  48.152 
(6)    6  =  3.5,  c  =  2.6,  ^  =  33°;    ^ns.  99°  58'.9,  47M'.l,  1.9356 
(c)    u=z22,    v  =  12,  ir=42°.     ^ns.  106°  27'.6,  31°  32'.4,  15.35 

3.  (a)    a  =  7,      b  =  12,     c  =  15  ;  ^ns.  27°  16',  51°  45'.2,  100°  58'.8 
(6)     i  =  10,  m  =  14,    n  =  20  ;  ^ns.  27°  39'. 6,  40°  32'.2,111°  48'.2 
(c)    u  =  3,      V  =  4,      ?/;  ==  5.      ^?is.  36°  52'. 2,  53°  7'.8,  90°  O'.O 

4.  (a)    a  =  50.8,  6  =  35.9,  u4  =  64°  ;  ^ns.  39°  26'.0,  76°34'.0,  54.973 

(6)    ,  =  6.22,  A.  =  7.48,  (?  =  26°  ;  ^r^.  j^'^'^'f '  ^''^  ^^ '^^  ^^'^^ 
^^  '  '  [148°ll'.l,  5°48'.9,  1.438 

(c)    b  =  23.4,  g  =  19.8,  B  =  109°  ;   Ans.  53°  8'.1,  17°  51'.9,  7.5922 

Id)    a  =  213,   b  =  278,    5  =  100°.    Ans.  48°  59'.2,  31°  0'.8,  145.45 

5.  To  determine  the  distance  from  a  point  A  to  an  inaccessible  object 
J5,  a  base  line  J.C  =  300  ft.  and  the  angles  BAG  =  40°,  BGA  =  50°  are 
measured.     Find  the  distance  AB.  Ans.  229.8 

6.  To  determine  the  distance  between  two  trees.  A,  B,  on  opposite  sides 
of  a  hill,  a  point  C  is  chosen  from  which  both  trees  are  visible;  the  dis- 
tances ^C  =  400  ft.,  BC  =  361  ft.,  and  the  angle  ACB  =  55°  are  then 
measured.     What  is  the  distance  between  the  trees  ?  Ans.  353.08 

7.  The  sides  of  a  triangular  field  are  43  rods,  48  rods,  and  57  rods, 
respectively  ;  determine  the  angles  between  the  sides. 

Ans.  47°  24',  55^  15',  77°  21'. 

8.  A  50-ft.  chord  of  a  circle  subtends  an  angle  of  100°  at  the  center. 
A  triangle  is  to  be  inscribed  in  the  larger  segment,  having  one  of  its  sides 
40  ft.  long.     How  long  is  the  other  side  ?    Is  there  only  one  solution  ? 

Ans.  65.22 

9.  A  triangle  having  one  of  its  sides  60  ft.  long  is  to  be  inscribed  in 
the  segment  of  Ex.  8.  Determine  the  remaining  side.  How  many  solu- 
tions are  there  in  this  case  ?  Ans.  18.88,  58.25 

10.  Find  the  length  of  a  side  of  an  e(iuilateral  triangle  circumscribed 
about  a  circle  of  radius  15  inches.  Ans.  51.96  in. 


48  PLANE  TRIGONOMETRY  [V,§34 

11.  The  angle  of  elevation  of  the  top  of  a  mountain  is  observed  at  a 
point  in  the  valley  to  be  60^  ;  on  going  directly  away  from  the  mountain 
one  half  mile  up  a  slope  inclined  30°  to  the  horizon,  the  angle  of  elevation 
of  the  top  is  found  to  be  20°.     Find  the  height  of  the  mountain. 

Ans.  4529.5  ft. 

12.  The  base  of  an  isosceles  triangle  is  245.5  and  each  of  the  base 
angles  is  68°  22^     Find  the  equal  sides  and  the  altitude. 

Ans.  332.96,  309.51 

13.  The  altitude  of  an  isosceles  triangle  is  32.2  and  each  of  the  base 
angles  is  32°  42'.     Find  the  sides  of  the  triangle.        Ans.  100.31,  59.60 

14.  A  chord  of  a  circle  is  100  ft.  long  and  subtends  an  angle  of  40°  42' 
at  the  center.     Find  the  radius  of  the  circle.  Ans.  143.78 

15.  From  a  point  directly  in  front  of  a  building  and  150  feet  away  from 
it,  the  length  of  the  building  subtends  an  angle  of  36°  44'.  How  long 
is  it  ?  Ans.  66.40 

16.  Find  the  perimeter  and  the  area  of  a  regular  pentagon  in- 
scribed in  a  circle  of  radius  12.  Ans.  70.534,  342.38 

17.  Find  the  perimeter  and  the  area  of  the  regular  octagon  formed  by 
cutting  off  the  corners  of  a  square  15  inches  on  a  side. 

Ans.  49.705,  186.39 

18.  Find  the  perimeter  and  the  area  of  a  regular  pentagon  whose 
diagonals  are  16.2  inches  long.  Ans.  50.06,  172.466 

19.  Find  the  perimeter  and  the  area  of  a  regular  dodecagon  inscribed 
in  a  circle  of  radius  24.  Ans.  149.08,  1728. 

20.  Two  chords  subtend  angles  of  72°  and  144°  respectively  at  the  center 
of  a  circle.  Show  that  when  they  are  parallel  and  on  the  same  side  of 
the  center,  the  distance  between  the  chords  is  one-half  the  radius. 

21.  Devise  a  formula  for  solving  an  isosceles  triangle  when  the  base 
and  the  base  angles  are  given  ;  when  the  base  and  one  of  the  equal  sides 
are  given ;  when  one  of  the  equal  sides  and  one  of  the  base  angles  are 
given. 


PART  n.     OBTUSE   ANGLES  AND    OBLIQUE 
TRIANGLES 


CHAPTER  VI 

FUNDAMENTAL  DEFINITIONS  AND   FORMULAS 

35.  Obtuse  Angles.  The  solution  of  oblique  triangles  in- 
volves obtuse  *  as  well  as  acute  angles.  For  this  reason  we 
need  to  be  able  to  determine  the  values  of  the  trigonometric 
ratios  for  such  angles  ;  it  is  not  necessary,  however,  to  enlarge 
our  tables  for  this  purpose,  for,  as  will  now  be  shown,  every  ratio 
for  an  obtuse  angle  can  he  expressed  in  terms  of  some  ratio  of  an 
acute  angle. 

Let  an  obtuse  angle  a  be  placed  on  the  coordinate  axes  with 
the  vertex  at  the  origin  and  one  side  along  the  ic-axis  to  the 
right ;  then  the  other  side  will  fall  in 
the  second  quadrant.  The  ratios  sin  a, 
cos  a,  etc.,  are  defined  in  terms  of  x,  y, 
and  r  =  Vx^  -\-  y^  precisely  as  they  were 
for  acute  angles  in  §  11.  It  should  be 
noticed,  however,  that  since  x  is  negative 
while  y  and  r  are  positive,  every  ratio 
which  involves  x  is  negative  for  an  obtuse  angle ;  thus  x/r  = 
cos  a,  y/x  =  tan  a,  and  their  reciprocals,  sec  a  and  ctn  a,  are 
all  negative  for  obtuse  angles. 

We  now  proceed  to  obtain  equations  similar  to  the  equations 
sin  (90°  —  a)  =  cos  a,  etc.  (proved  in  §  15),  which  enabled  us 
to  find  the  values  of  the  ratios  of  acute  angles  greater  than 
45°  in  terms  of  the  ratios  of  angles  less  than  45°. 

*  An  obtuse  angle  is  an  angle  which  is  greater  than  90°  and  less  than  180°. 
B  49 


>x 


50 


PLANE  TRIGONOMETRY 


[VI,  §  36 


36.  Reduction  from  Obtuse  to  Acute  Angles.    Let  a  be 

placed  on  coordinate  axes  as  described  above,  and  let  the 
supplement  of  a  be  denoted  by  /3  (which  is  an  acute  angle). 
Lay  off  j8  in  the  first  quadrant  with  one  side  along  the  aj-axis. 
From  a  point  P  in  the  side  of  a  (in  second  quadrant)  and 
a  point  P  in  the  side  of  /3  (in  first  quadrant)  at  the  same  dis- 
tance r  from  the  origin,  draw  the 
perpendiculars  FM,  F'M',  as  in 
Eig.  41.  The  value  of  x  for  the 
point  F  will  be  negative  since  F  is 
in  the  second  quadrant.  Let  its  co- 
ordinates be  (—a,  b)  ;  then,  since 
the  triangles  OFM,  OFM'  are 
symmetric,  the  coordinates  of  F  are  (a,  h) .    As  in  §  11,  we  have 


ky 


(-a,  b) 


\^ 


M' 


(a.  b) 
b 


Fig.  41. 


sin  a  =  -  =  sin  p, 
r 


:180° 


cos  a  =  - 


a 
r 


-cos  ^, 


or,  since  /3  - 

(1)  sin  a  =  sin  (180°  -  a)  ; 

(2)  cos  a  =  -  cos  (180°  -  a) . 
In  a  similar  manner  it  can  be  shown  that 

(3)  tan  a  =  --  tan  (180°  -  a). 

It  follows  that  if  a  is  an  obtuse  angle  we  find  its  sine  by 
looking  for  the  sine  of  its  supplement,  which  is  an  acute  angle,  and 
similarly  for  the  other  functions,  always  having  regard  for  the 
proper  sign. 


EXERCISES  XII. 


-FUNCTIONS  OF  OBTUSE  ANGLES 

AY 


1.   From  the  accompanying  fig 
the  following  relations: 

(a)  sin  (90*^  +  a)  =  cos  a. 
(6)  cos  (90°  +  a)  =  -  sin  a. 
(c)   tan  (90°  -\-a)  =-  ctn  a. 
{(i)  ctn  (90°  +  a)  =  -  tan  a. 


ure  prove  p 


.?o' 


p' 


Fig.  42. 


VI,  §  38]  LAW  OF  COSINES  51 

2.  Construct  obtuse  angles  whose  functions  have  the  following  values  : 

(a)  sin  e  =  1/3.         (5)  tan  (9  =  -  3/4.         (c)  cos  ^  =  -  3/.5. 
(c!)  sin  e  =  1/2.         (e)  sin  0  =  V2/2.  (/)  sin  0  =  V3/2. 

3.  Find  the  values  of  the  remaining  functions  of  the  angles  of  Ex.  2. 

4.  Express  the  following  as  functions  of  an  angle  less  than  45°,  and 
look  up  their  values  in  a  table. 

(a)  sin  121°.  (6)  cos  101°.                   (c)  tan  168°. 

(d)  sin  99°.  (e)  ctn  178°.                  (/)  cos  154°. 

(g)  cos  133°  11'.  (h)  tan  144°  38'.            (i)  sin  92°  3'. 

5.  Solve  the  equation  6  cos2  x  +  7  cos  ic  +  2  =  0. 

[To  solve  an  equation  of  this  type  one  should  first  regard  it  as  an 
algebraic  (quadratic)  equation  in  which  the  unknown  is  cos  x  :  replacing 
cos  X  by  the  letter  t  we  have  the  equation  6^24.7^4.2  =  0.  The  solu- 
tions of  this  equation  are  i  (or  cos  x)  =  —  1  or  ^  =  —  |.  Then  find 
from  the  tables  the  angles  x  satisfying  the  equations  cos  x  =  —  J  and 
cos  X  =  —  I ;  they  are  x  =  120°  or  x  =  131°  48'. 6] 

6.  Show  that  the  equation  tan  x  =  c  has  an  obtuse  angle  solution  if  c 
is  any  negative  number. 

7.  Show  that  the  equation  sin  jc  =  c  has  both  an  acute  and  an  obtuse 
angle  solution  if  c  is  any  positive  number  less  than  1. 

8.  Show  that  the  equation  cos  x  =  c  has  a  solution  between  0°  and 
180°  if  c  Hes  between  +  1  and  —  1,  and  that  this  solution  is  an  acute 
angle  if  c  is  positive  and  an  obtuse  angle  if  c  is  negative. 

9.  Find  all  of  the  solutions  between  0°  and  180°  for  the  following 
equations  : 

(a)  3  sin2  x  -  2  sin  x  —  1  =  0.  (6)  4  sin2  x  —  3  sin  x  -  1  =  0. 

(c)  6  sin2  X  +  sin  X  —  1  =  0.  (d)  6  sin2  x  —  sin  x  -  1  =  0. 

37.  Geometric  Relations.  In  the  following  sections  certain 
fundamental  geometric  and  trigonometric  relations  connecting 
the  sides  and  angles  of  any  triangle  are  given.  Upon  these  is 
based  a  systematic  method  of  solution  of  oblique  triangles, 
which  is  given  in  the  following  chapter. 

38.  The  Law  of  Cosines.  In  any  triangle,  the  square  of  any 
side  is  equal  to  the  sum  of  the  squares  of  the  other  two  sides  minus 
twice  their  product  into  the  cosine  of  their  included  angle. 

Denote  the  sides  of  a  triangle  by  a,  6,  c,  and  the  angles 
opposite  by  A,  B,  C  ]  and  express  the  square  of  side  a  in  terms 


52 


PLANE  TRIGONOMETRY 


[VI,  §38 


of  by  c,  and  C  as  follows.     Drop  a  perpendicular,  p,  from  B  to 
the  opposite  side  and  denote  the  segments  of  this  side  by  x 


and  y.     By  (13,  14)  §  12,  we  have  in  Fig.  43, 

p  =  c  sin  A,      X—  c  cos  A,      y  =  b  —  x=b  —  c  cos  A 
a^  =  y^^p^=  (b  —c  cos  Ay  +  c2  sin2  ^ 
=  62  _  2  &c  cos  ^  4-  c2  (cos2  ^  +  sin2  A) 
whence,  since  sin^  A  +  cos^  A  =  l 
(4)  a2  =  62  _^  c2  —  2  &c  cos  -4. 

If  as  in  Fig.  44  the  side  a  to  be  found  is  opposite  an  obtuse 
angle  A,  y=b  +  x',  but  by  (2)  §  36,  x=c  cos  (180°-^)  = 
—  c  cos  A ;  hence  y  =b  —  c  cos  A  and  p=  c  sin  (180°  —  ^)  = 
c  sin  J[,  exactly  as  in  the  case  considered  above. 

The  law  of  cosines  can  be  used  to  compute  one  side  of  a 
triangle  when  the  other  two  sides  and  one  angle  are  known, 
and  also  to  find  the  angles  when  the  three  sides  are  known. 

Example  1.  One  angle  of  a  triangle  is  66°  25'  and  the  including  sides 
are  3  and  6.     Find  the  third  side. 

ic2  =  32  +  52  -  30  (.4)  =22,  ,'.x  =  a/22  =  4.69 

Example  2.  Two  sides  of  a  triangle  are  7  and  8  and  the  angle  opposite 
the  former  is  60°.     Find  the  third  side. 

72  =  x2  +  82-16x  (J) 
whence  «  =  3  or  «  =  6  and  there  are  two  solutions. 

Example  3.  The  sides  of  a  triangle  are  3,  5,  and  7.  Find  the  greatest 
angle. 

72  =  32  +  62-30  cos  X 

whence  cos  x  =  —  \  and  x  =  120°. 


VI,  §39] 


LAW  OF  SINES 


53 


EXERCISES  XIII.  — THE  COSINE  LAW 

1.  Two  sides  of  a  triangle  are  1.5  and  2.4,  and  their  included  angle  is 
36°.     Find  the  third  side.  Arts.  1.48 

2.  Two  sides  of  a  triangle  are  5  and  8  and  the  included  angle  is  135°. 
Find  the  third  side.  Arts.  5.69 

3.  Two  sides  of  a  triangle  are  3  and  4  and  the  angle  opposite  the 
former  is  30°.     Find  the  tliird  side.  Arts.  2  V3  +  V5  or  2  V3  -  V5. 

4.  The  sides  of  a  triangle  are  3,  5,  and  6.     Find  the  smallest  angle. 

Ans,  29°  55'.6 

5.  The  sides  of  a  triangle  are  10,  14,  and  17.     Find  the  angles. 

Ans,  36°  1',  55°  25',  88°  34'. 

* 

6.  Two  sides  of  a  triangle  are  11  and  17,  and  the  angle  opposite  the 
former  is  30°.     Find  the  third  side  by  the  law  of  cosines. 

7.  Devise  a  method  for  finding  the  angle  between  two  lines  without 
an  instrument  for  measuring  angles.  Could  the  law  of  cosines  be  used 
for  this  purpose  ? 

39.  The  Law  of  Sines.  Any  two  sides  of  a  triangle  are  to 
each  other  as  the  sines  of  the  angles  opposite. 

Denote  the  sides  and  angles  of  a  triangle  by  a,  &,  c,  A,  B,  C, 

as  above.     Prove  that 

a  __  sin  A 

6      sin  5 
as  follows : 

Drop  a  perpendicular  from  C  (the  angle  included  by  the 
sides  a  and  b)  to  the  opposite  side.     In  Fig.  45,  where  the 


Fig.  45.  Fig.  46. 

angles  A  and  B  are  both  acute,  by  (13),  §  12 

p  =  a  sin  B  and  also  p  =  b  sin  A, 
whence  a  sin  B  =  b  sin  A 


54  PLANE  TRIGONOMETRY  [VI,  §  40 

and 

(5) 


and  dividing  through  by  h  sin  B, 

a      sin^ 


h      sin  JB 

In  Eig.  46,  where  one  of  the  given  angles  is  obtuse, 
p  =  a  sin  5'  =  a  sin  (180°  —  B)=  a  sin  B 
and  also  p  =  b  sin  A,  exactly  as  above. 

If  the  perpendicular  is  drawn  from  one  of  the  other  vertices, 
say  from  A,  the  above  procedure  leads  to 

^  ^  c^sinC' 

Erom  equation  (5),  dividing  each  side  by  sin  A  and  multi- 
plying each  side  by  6,  we  see  that 

g     __     b 
sin  A     sin  B 
Erom  (6)  we  see,  similarly,  that  each  of  these  ratios  is  equal 
to  c/sin  (7.     It  follows  that  we  have 

a  b  c 


(7) 


sin  A      sin  B      sin  C 


40.  Diameter  of  Circumscribed  Circle,    it  can  be  shown  that 

each  of  the  ratios  in  (7)  (where  a,  6,  c,  stand  for  the  numerical  measures 
of  the  sides)  is  equal  to  the  numerical  measure  of  the  diameter  of  the  cir- 
cumscribed circle  ;  and  this  furnishes  another  proof  of  the  law  of  sines. 

Circumscribe  a  circle  about  the  triangle 
ABC,  draw  the  diameter  BA'=  d,  and  con- 
nect A'  C.  Then  angle  A'  CB  is  a  right  angle 
and  A'=A  since  each  is  measured  by  one-half 
the  arc  BC,  Therefore  by  (16),  §  12, 
a  a 


B 


sin  A'  sin  A 

b  c 

and  similarly  d  = ,  d  = 

^         sinB'  sinC 


Fig.  47. 


If  the  angle  A  were  obtuse  we  should  have 
A'  =  180°  -  A,  but  since  sin  (180°  —  A')  = 
sin  A,  the  same  result  holds  in  this  case  also.  Therefore  in  general,  the 
diameter  of  the  circle  circumscribed  about  a  triangle  is  equal  to  any 
side  divided  by  the  sine  of  the  opposite  angle. 


VI,  §  41]  LAW  OF  SINES  55 

The  law  of  sines  can  be  used  whenever  three  parts  of  a  tri- 
angle are  known,  of  which  two  are  a  side  and  the  angle 
opposite. 

Example  1.  Two  angles  of  a  triangle  are  10°  12'  and  46°  36'  and  the 
shortest  side  is  10.     Find  the  longest  side. 

The  angle  opposite  the  longest  side  is  123°  12'  and 

g  ^        10 

sin  123°  12'      sin  10°  12' 

whence  ^  ^  10(.83676)  ^ 

.17708 

Example  2.  The  three  sides  of  a  triangle  are  3,  5,  and  7.  We  have 
seen  in  Ex.  3,  p.  62,  that  the  largest  angle  is  120"'.  Find  the  smallest 
angle. 


sin  X     sin  120° 

whence  sinx  =?^=  .37115 

14 
and,  since  x  must  be  acute, 

x  =  21°47'.2 

EXERCISES  XIV.  — THE  SINE  LAW 

1.  Two  angles  of  a  triangle  are  19°  and  104°  and  the  side  opposite  the 
former  is  20.     Find  the  other  two  sides.  Ans,    61.5,  59.6 

2.  The  sides  of  a  triangle  are  8,  13,  and  15.  Find  the  angle  opposite 
the  second  side  by  the  law  of  cosines  and  the  other  two  by  the  law  of 
sines.  Ans.    60°,  32°  12',  87°  48  . 

3.  The  sides  of  a  triangle  are  21,  26,  31.     Find  the  angles  as  in  Ex.  2. 

Ans.    56°  7',  42°  6',  81°  47'. 

4.  Compute  the  length  of  the  radius  of  the  circumscribed  circle  for 
each  of  the  triangles  in  Exs.  1-3. 

5.  Two  angles  of  a  triangle  are  38°  12'  and  61°  10',  and  the  included 
side  is  350.6    Find  the  other  two  sides.  Ans.   219.7,311.3 

41.  The  Law  of  Tangents.  In  any  triangle  the  difference  of 
any  two  sides  is  to  their  sum  as  the  tangent  of  one-half  the  differ- 
ence of  the  angles  opposite  those  sides  is  to  the  tangent  of  one-half 
their  sum. 

Let  ABC  be  any  triangle  having  two  sides  a  and  b  unequal, 
say  a>b;  the  included  angle  C  may  be  acute,  right,  or  obtuse. 


56 


PLANE  TRIGONOMETRY 


IVI,  §  41 


With  a  radius  bj  the  shorter  of  the  given  sides,  and  center  (7, 
the  vertex  of  the  included  angle,  describe  a  circle  through  A 


k a-b H 


Fig.  48. 


which  cuts  the  side  CB  in  a  point  Z)  between  B  and  C  and 
also  at  a  second  point  E  beyond  C.  Draw  EA,  and  at  B  erect 
a  perpendicular  which  meets  i^^l  produced  at  F,  On  Di^  as  a 
diameter  construct  a  circle  ;  this  circle  will  pass  through  A  and 
By  for  FAD  is  a  right  angle  since  it  is  the  supplement  of  DAE 
which  is  inscribed  in  a  semicircle,  and  FBD  is  sl  right  angle 
by  construction.  This  construction  is  possible  for  any  triangle 
in  which  a>b. 

Angle  BFE  =  ^(A  +  B)  since  it  is  the  complement  of  angle 
CEA  =  i  (7;  and  -i ^  +  i  J5  +  i  C=  90°  since  the  sum  of  the 
angles  of  a  triangle  is  180°.  Angle  DFA  =  B  since  each  is 
measured  by  one-half  the  arc  AD ;  therefore  BFD  =  BFE  — 
DFA  =  ^(A  +  B)-B=i{A-B). 

In  the  right  triangles  DBF  smd  EBFhj  (13),  §  12, 

a  -  6  =  J5i^.  tan  ^{A  -  B), 

a  +  b  =  BF'  tan  ^(A  +  B), 


whence 
(8) 


a_&^tan|(>t-g) 
a  +  b     tani(A  +  B)' 


VI,  §  42] 


LAW  OF  TANGENTS 


57 


This  formula  is  still  true  but  trivial,  if  a  =  b,  since  in  that 
case  each  side  reduces  to  zero ;  if  a  <  6,  the  result  would  ob- 
viously be  * 
/Q\                          b  —  a  __  tan  ^(B  —  A)  ^ 
^  ^                          6+~a""tani(B-+-^) 

Since  ^(A  -{-  B)  is  the  complement  of  i  (7,  (8)  can  be  re- 
duced to  the  form 


(10) 


t^nUA^B)- 


a-i-b 


ctn  J  C. 


42.  Tangents  of  the  Half-angles.  The  tangent  of  one-half 
any  angle  of  a  triangle  can  be  expressed  in  terms  of  the  sides 
as  follows. 

Bisect  the  angles  of  the  triangle  ABC  and  draw  the  in- 
scribed circle  tangent  to  the  sides  at  P,  Q,  and  B.  Let  r  be 
the  radius  of  this  circle  and  let  8  stand  for  one-half  the  perime- 
ter of  the  given  triangle,  i.e. 

2s  =  a  +  b-{'C. 
Then 

AP=AB,  BE  =  BQ,    CQ=CF, 
and 

BE-^BQ+CQ+CF  =  2BQ-{-2QC=2a, 

whence 

2AP=2s-2a 
and 

AP=AE  =  s-a, 
Similarly, 

BE  =  BQ  =  s--b 
and 

(7Q=CP  =  s-c. 

In  the  right  triangle  AFO,  by  (12), 

§12, 

tan  i  A= 

s—  a 

A  similar  result  holds  for  the  other  two  angles.     Hence  we 

have  the  three  formulas  : 


Fig.  49. 


(11)     tanl>l  = 


tani5  =  . 


s  —  a 


-b' 


tanlC=-    ' 


5  — C 


68 


PLANE  TRIGONOMETRY 


[VI,  §  43 


Fia.  50. 


43.   Radius  of  the  Inscribed  Circle.     It  remains  to  express 
r  in  terms  of  the  sides  of  the  triangle.     In  the  triangle  ABG 

produce  the  sides  AB  and 
AO,  Bisect  the  angle  A 
and  the  exterior  angles  at 
B  and  (7.  These  bisectors 
meet  in  a  point  /  which  is 
the  center  of  a  circle  which 
touches  the  side  a,  and  the 
sides  h  and  c  produced. 
This  circle  is  called  an 
escribed  circle  of  the  triangle.  Denote  its  radius  by  r'  and 
mark  the  points  of  tangency  P,  Q,  R.     Then  we  have 

Aq  =  AP,  BQ=BB,    CP=CB, 

therefore 

AB  +  BE  =  AC+CB  =  s, 

where  s  denotes  half  the  perimeter  of  the  given  triangle.     It 
follows  that  AQ  =  s  and 

BQ^AQ-^AB^S'-c. 

In  the  right  triangle  BQI, 

angle  IBQ  =  1(180°  -  B)=  90°  -  i  5 


and  therefore  angle  BIQ  - 
§  42,  in  triangle  BQI, 


r  B ;  then  by  (13, 14),  §  12  and  (11), 


:(s-c)ctniB  =  ^'-^^^'~'\ 


and  in  triangle  AQI, 


r^  =  s  tan  ^  A  =  - 


Equating  these  two  values  of  r'  and  solving  for  r,  we  have, 
(12)  •         r=-\/5E^(fEMEf). 

The  symmetry  of  this  result  in  a,  &,  c  shows  that  we  shall 
get  the  same  result  if  we  produce  sides  c  and  a^  or  a  and  &, 


VI,  §  43]  HALF  ANGLE  FORMULAS  59 

Example  1.     The  sides  of  a  triangle  are  145/13,  119/13,  and  156/13. 
Find  the  radius  of  the  inscribed  circle  and  the  angles  of  the  triangle. 
We  first  compute  the  values  of  s,  s  —  a,  s  —  6,  and  s  —  c 

s  =  1  (a  +  6  +  c)  =  210/13,    s-a  =  65/13  =  5,    s  -  6  =  7,  s  -  c  =  54/13. 

Substituting  in  the  formula  for  r  we  obtain 


^        /7x5x(54/13)^^^^ 
^  210/13 


tani^=— ?^  =  3/5,  tan*  5  =-^^  =  3/7,   tan  1  C  = —^  =  13/18  ; 
s  —  a  s  —  b  s  —  c 

hence  from  the  tables  we  find 

^/2  =  30°57^8,    i?/2  =  23°  11^9,    C/2  =  35°50'.3 

Example  2.  Two  sides  of  a  triangle  are  12  and  8  and  the  included 
angle  is  60^.     Find  the  remaining  angles. 

Denoting  the  unknown  angles  by  A  and  B  we  have 

A+  B  =  180°  -  60°  =  120°, 

then  by  the  law  of  tangents  we  have 

12  -  8  ^  tan  K^  -  -B)  _  tan  ^(A  -  JB) 
12  +  8  tan  60°        ~  V3  ' 

hence 

tan  l(A  -  B)  =  V3/5  and  }  (A-  B)=  19°  6'.4 

Adding  this  result  to  i(A  +  5)  =  60°  we  obtain  A  =  79°6'.4,  and  sub- 
tracting we  get  B  =  40°  53'.6 

EXERCISES 

1.  The  three  sides  of  a  triangle  are  7,  12,  and  15.  Find  the  radius  of 
the  inscribed  circle  and  thQ  angles. 

2.  Determine  the  angles  of  the  following  triangles  : 

(a)  a  =  5,     6  =  9,     c  =  11.  (c)   a  =  10,  6  =  12,  c  =  15. 

(6)  a  =  4,     6  =  8,     c  =  10.  (d)  a  =  6,     6  =  8,     c  =  10. 

3.  Determine  the  angles  and  third  side  of  the  following  triangles  : 
(a)  a  =  4,     6  =  8,     C  =  20°.  (c)  a  =  10,  6  =  12,   G  =  35°. 
(6)  a  =  4,     6  =  8,     C  =  40°.  (d)  a  =  13,  6  =  17,   C  =  44°. 

4.  To  determine  the  distance  between  two  objects  A  and  B  separated 
by  a  barrier,  the  distances  J.0  =  40  rd.,  BC  =  4S  rd.  are  measured  to  a 
third  point  C  The  angle  ACB  =  68°  is  then  measured.  Find  the  dis- 
tance AB  and  the  other  angles  of  the  triangle  ABC. 


CHAPTER  VII 
SYSTEMATIC   SOLUTION   OF   OBLIQUE   TRIANGLES 

44.  Analysis  of  Data.  In  the  solution  of  oblique  triangles 
the  following  cases  arise  : 

Case  I.  Given  two  angles  and  a  side. 

Case  n.  Given  two  sides  and  the  included  angle. 

Case  III.  Given  the  three  sides. 

Case  IV.  Given  two  sides  and  an  angle  opposite  one  of  them. 

The  direction  '•'■Solve  a  triangle^''  tacitly  assumes  that  a  suf&cient 
number  of  parts  of  an  actual  triangle  are  given.  A  proposed  problem 
may  violate  this  assumption  and  there  v^^ill  be  no  solution.  Thus  there  is 
no  triangle  whose  sides  are  14,  24,  and  40.  An  attempt  to  solve  such  an 
impossible  problem  gives  rise  to  a  contradiction  such  as,  for  example,  the 
sine  or  cosine  of  some  angle  greater  than  1.  Any  triangle  which  can  be 
constructed  can  be  solved. 

45.  Case  I.  Given  Two  Angles  and  a  Side.  In  this  case 
it  is  immaterial  which  side  is  given,  since  the  third  angle  can 
be  found  from  the  fact  that  the  sum  of  the  three  angles  is  180°. 

There  is  one  and  only  one  solution,  provided  the  sum  of  the 
given  angles  is  less  than  180°. 

TJie  other  two  sides  can  be  found,  one  at  a  time,  by  the  law  of 
sines  (§  39). 

Example  1.     Given  one  side  of  a  triangle  a  =  2.903  and  two  of  the 
Q  angles  B  =  79°  40^  C  =  33^  15'  ;  find  the 

remaining  parts. 

A  =  180°  -  (79°  40'  +  33°  16')  =  67°  6'. 
By  the  law  of  sines 

b     _  sin  79°  40^ 
Fig.  51.  2.903      sin  67°  6'' 

60 


VII,  §  45]    SOLUTION  OF  OBLIQUE  TRIANGLES  61 

Many  of  the  computations  in  the  solution  of  triangles  are  of  the  follow- 
ing type.      To  find  one  term  of  a  proportion,  -  =  - ,  when  the  other  three 

b  d 
are  known,  no  matter  in  which  of  the  four  positions  the  unknown  stands. 
The  student  should  master  this  problem.  The  following  rule  applies. 
Imagine  the  means,  and  also  the  extremes,  to  be  connected  by  straight  lines 
crossing  at  the  =  sign.  Multiply  together  the  pair  of  knowns  thus  con- 
nected and  divide  by  the  known  opposite  the  unknown. 

Applying  this  rule  to  the  computation  of  b,  the  work  may  be  written 
down  as  follows : 


sm  79°  40'  =  .98378 

sin  67°  5'  =  .92107)2.85691334  |3.1007 

2.903 

2  76321 

295134 

92703 

885402 

92107 

196756 

59634 

2.85591334 

.-.  6  =  3.1007 

This  work  can  be  shortened  by  the  use  of  logarithms.  In 
all  cases  where  the  product  of  two  or  more  numbers  is  to  be 
divided  by  other  numbers  we  can  use  the  following  principle 
(Tables,  p.  x).  Subtracting  the  logarithm  of  a  number  is  equivor 
lent  to  adding  its  cologarithm. 

The  computation  of  5  by  logarithms  may  be  written  as  follotvs  : 
log  2.903  =  0.46285 
log  sin  79°  40'  =  9.99290  -  10 
colog  sin  67°  6'  =  0.03571 
log  b  =  0.49146 
6  =  3.1007 
The  side  c  is  found  similarly  from  the  proportion 

c     ^  sin  33°  15' 
2.903      sin  67°  5'* 

To  check,  apply  the  law  of  sines  (§  39),  or  the  Law  of 
tangents  (§  41)  to  the  computed  sides  b  and  c. 

EXERCISES  XV.  — CASE  I 

Solve  the  following  triangles.  SmaU  letters  represent  sides  and  cor- 
responding capital  letters  the  angles  opposite. 

1.  B  =  50°  30',        C  =  122°  9',        a  =  72.        Ans.   334.28,  476.51 

2.  F  =  82°20',        G«  =  43°20',         /=48.        Ans.    33.097,39.165 

3.  M  =  79°  59',       iV^  =  44°41',        p  =  477.      Ans,   340.73,  398.39 


4. 

P  =  37°  58', 

5. 

A  =  70°  55', 

6. 

A=  51°  47', 

7. 

A  =  48°  10', 

8. 

B  =  38°  12', 

9. 

Z7=46°36', 

10. 

B  =  21°  16', 

11. 

B  =  62°  42', 

12. 

B  =  58°20', 

13. 

G  =  43°  50'.4, 

14. 

G^  =  75°2'.7, 

15. 

Two  observers 

62  PLANE  TRIGONOMETRY  [VII,  §  46 

Q  =  65°2',  r  =  133.2    Ans.    84.103,110.679 

ir=:52°9',  a  =  48.09    Ans.    42.645,  40.031 

5  =  66°  20',  c  =  337.6  Ans.  300.73,350.58 
5  =  54°  10',  c  =  38.7  ^ns.  29.516,  32.116 
C  =  61°10',  a  =  70.12  Ans.  43.949,62.257 
F=124°18',  w;  =  1001.  Ans.  4598.6,5228.4 
C=113°34',  d  =  20.93  Ans.  10.705,27.053 
ikf=52°22',  a  =  39.75  Ans.  38.995,34.753 
Gf  =  61°2'.3,  ^  =  8.75  Ans.  8.512,  8.715 
Q  =  69°30'.2,  c  =  73.05  Ans.  96.685,97.123 
ir=43°44'.3.  A:  =  81.5  Ans.  103.32,113.89 
Two  observers,  facing  each  other  3  kilometers  apart  and  at  the 
same  altitude,  find  the  angles  of  elevation  of  a  Zeppelin  to  be  57°  20'  and 
64°  30',  respectively.     Find  the  height.  Ans.    2.683 

16.  A  diagonal  of  a  parallelogram  is  18.56  and  it  makes  angles  26°  30' 
and  38°  40'  with  the  sides.  Find  the  sides  and  the  area  of  the  parallelo- 
gram. Ans.   9.125,  12.777,  105.81 

17.  A  lighthouse  was  observed  from  a  ship  to  be  N.  16°  W. ;  after  sail- 
ing due  east  4.5  miles,  the  lighthouse  was  N.  48°  W.  Find  the  distance 
from  the  lighthouse  to  the  ship  in  both  positions.        Ans.   5.682,  8.163 

18.  The  side  of  a  hill  is  inclined  at  an  angle  of  22°  37'  to  the  horizon. 
A  flagstaff  at  the  top  of  the  hill  subtends  an  angle  of  13°  17'  from  a  point 
at  the  foot  of  the  hill,  and  an  angle  of  18°  2'  from  a  point  100  ft.  directly 
up  the  hill.     Find  the  height  of  the  flagstaff.  Ans.   95.053 

19.  To  find  the  distance  from  a  station  A  to  an  inaccessible  point  5,  a 
base  fine  AC  =  600  ft.,  and  the  angles  ACS  =  68°  18',  CAB  =  58°  28'  are 
measured.     Find  the  distance  AB, 

20.  To  find  the  height  of  an  inaccessible  object  AB,  a  base  line  CD  = 
250  ft.  is  measured  directly  toward  the  object :  also  the  angles  of  eleva- 
tion ADB  =  48° 20'  and  ACB  =  38°  40'.     Find  the  height  AB. 

46.   Case  II.    Given  Two  Sides  and  the  Included  Angle. 

There  is  always  one  and  only  one  solution. 

The  obvious  method  of  solution  is  to  find  the  third  side  by 
the  law  of  cosines  (§  38),  and  then  the  other  two  angles  by 
the  law  of  sines  (§  39). 

Example  1.  Two  sides  of  a  triangle  are  10  and  11,  and  the  included 
angle  is  36°  24' .     Find  the  other  parts. 


VII,  §  47]     SOLUTION  OF  OBLIQUE  TRIANGLES 


63 


Draw  a  figure,  denote  the  unknown  side  by  a,  and  the  unknown 
angles  by  jB,  G.    Then  we  may  write 

a2  =  10^  +  TT^  _  2(10)  (11)  cos  35°  24', 

a2=221-(220)(.81513). 
Then    a2  _  41.6714,     whence     a  =  6.4553 
(Tables,  p.  104) .  ^ 

To  find  B  and  C  by  the  law  of  sines,  we 
have 


sin  B 


11 


and 


sin  G 


10 


sin  35^  24'      6.4553'  sin  35°  24'      6.4553' 

whence  on  computing  (see  Example  1,  §  45) 

5=80^47'.0,         C=63°48'.8 
Check  :  A  +  B -{-  G  =  179""  59'. 8 

Example  2.  Two  sides  of  a  triangle  are  138.65 
and  226.19,  and  the  included  angle  is  69°  12'. 9. 
Find  the  third  side. 

Construct  the  triangle  as  in  Fig.  53. 

a2  =  138-652  +  226.19^ 

-  2(138.65) (226.19)  cos  59°  12'.9 

While  this  is  not  adapted  to  logarithms,  neverthe- 
less logarithms  can  be  used  to  compute  separately 
the  three  terms  on  the  right ;  for  the  moment  call 
the  third  term,  x. 


log  138.65  =2.14192 

2 

4.28384 

(138.65)2  =     19224 

51161 

70385 

x=     32102 

a2  =     38283 

a  =  195.66 

(Tables,  p.  95) 


log226.19  =  2.35447 

2 


(226.19)2 


4.70894 
:  51161 


log  2  =  0.30103 

log  cos  59°  12'.9  =  9.70911 

2.14192 

2.35447 

lOgX  : 


;  4.50653 
X  =  32102 


47.  Logarithmic  Solution  of  Case  11.  When  two  sides  and 
•the  included  angle  are  given,  a  triangle  can  be  completely 
solved  by  logarithms  by  finding  first  the  other  two  angles  by 
the  law  of  tangents  (§  41). 


64 


PLANE  TRIGONOMETRY 


[VII,  §  47 


Example  1.  In  a  triangle  MPT,  side  m  =  138.65, 
side  t  =  226.19,  and  the  included  angle  P  =  69°  12^9. 
rind  the  other  parts. 

Applying  the  law  of  tangents  to  the  given  sides, 
noting  that  i>m, 

t-m_t2ini(T-M) 
t  +  m     tanj(r+^* 

In  this  proportion  three  terms  are  known  since 
T-\-  M=  180°  —  P.  The  work  may  be  set  down  as 
follows. 


t  =  226.19 
m  =  138.65 
«~m=    87.54 
t  +  m  =  364.84 
i(T+M)=  J(180°  -  P)  =  60° 23'.55 
i(T-M)  =  22°  53^5 

.-.  r=  83°  17' 

^1^=  37°  30' 


log  (i-m)=  1.94221 
colog  lt  +  m)=  7.43790  -  10 
log  tan  i(T+  M)  =  0.24646 
log  tan  i(  T  -  if)  =  9.62567  -  10 


The  side  p  can  now  be  found  by  solving  the  proportion 

p      ^sin69°12'.9 
138.66       sin  37°  30' 
log  138.66  =  2.14192 
log  sin  69°  12'.9  =  9.93404  ~  10 
colog  sin  37°  30'  =  0.21666 
logp  =r  2.29161 

from  which  p  =  196.66      Compare  Example  2,  §  46. 


1. 

(a) 

W 
(O 
(d) 

(O 
(/) 
(9) 
(7i) 

2. 

53°  8'. 


EXERCISES  XVI.  — CASE  II 

Solve  the  following  triangles  by  using  the  law  of  cosines  : 

a  =  22,      6  =  12,    O  =  42°.        Arts.    106°  27'.7,  31°  32'.4,  16.35 
5  =  62°.  Ans.    66°  13',  71°  47',  13.27 

iV^=126°.        Ans.   23° 46' .6,  31°  13'.4,  66.89 


a  =  14, 
I  =28, 
a  =  21, 
a  =  2.2, 
I  =13, 

M  =  41, 

5  =  3.5, 


c  =  16, 
m  =  36, 

6=24, 

h  =  4.2, 
m  =  16, 

tj  =  51. 


c  =  28.  Ans.  46°61'.6,  66°30'.3,  76°38'.l 
c  =  5.5  Ans.  21°16'.9,  43°61'.4,  114°61'.7 
w  =  20.  Ans.  40°27'.l,  62°69'.6,  86°33'.3 
W=61°,  Ans.   69°67\3,  49°  2'.7,  47.48 

c  =  2.6,  ^  =  33°.  Ans.   47°1'.3,  99°58'.7,  1.935 


Two  sides  of  a  triangle  are  2.1  and  3.5  and  the  included  angle  is 
Determine  the  remaining  parts.  Ans.   36°  52',  90°,  2.4 


VII,  §48]     SOLUTION  OF  OBLIQUE  TRIANGLES  65 

3. .  How  long  is  a  rod  which  subtends  an  angle  of  60°  at  a  point  which 
is  6  ft.  from  one  end  of  the  rod  and  8  ft.  from  the  other  ?       Arts.   7  ft. 

4.  How  long  is  a  rod  which  subtends  an  angle  of  120°  at  a  point  3  ft. 
from  one  end  and  5  ft.  from  the  other  ?  Ans,   7  ft. 

5.  Solve  each  of  the  following  triangles,  using  logarithms  : 

(a)  a  =  52.8,       6  =  25.2,       0=124°  34'.       ^ns.    17°  IIM,  70.233 

(6)6  =  55.1,       c  =  45.2,      ^  =  16°  16'.         ^ns.   47°  14'.1,  17.246 

(c)i=131,       m  =  n,         JV=39°46'.         ^ns.    31°  19'. 9,  88.568 

(d  )  a  =  35,  6  =  21,  0  =  48°  48'.         Ans.   36°  44'.4,  26.415 

(e)  u  =  604,  V  =  291,  W=  106°  19'.  Ans.  22°  9'.5,  740.45 
(/)  a  =  23.45,     6  =  18.44,    D  =  81°  50'. 

Ans.  56°56'.4,  41°13'.6,  27.696 
(  gr)  u  =  .6238,     V  =  .2347,     C  =  108°  30'. 

Ans.    53°49'.2,  17°40'.8,  0.7329 

6.  Two  sides  of  a  triangle  are  22.531  and  34.645  ;•  the  included  angle 
is  43°  31'.     Determine  the  remaining  parts. 

Ans.   40°16'.7,  96°  12'.3,   23.716 

7.  To  determine  the  distance  between  two  objects  A  and  B  separated 
by  a  hill,  the  distances  J. C  =  300  ft.,  BC  =  277  ft.,  and  the  angle  ACB 
=  65°  47',  are  measured.  From  these  measurements  find  the  distance 
AB.  Ans.   313.94 

8.  Two  objects,  A^  B,  are  separated  by  an  impassable  swamp.  A 
station  C  is  selected  from  which  distances  in  a  straight  line  can  be  meas- 
ured to  each  of  the  objects.  These  distances  are  found  to  be  CA  =  341 
ft.  7  in.,  CB  =  237  ft.  5  in.,  and  the  angle  ACB  is  found  to  be  53°  11'. 
Find  the  distance  AB.  Ans.   275.4 

9.  Two  objects,  J.,  B^  are  separated  by  a  building.  To  determine 
the  direction  of  the  line  joining  them,  a  point  C  is  taken  from  which  both 
A  and  B  are  visible  and  the  distances  J. C  =  200  ft.,  BC  =  137  ft.  9  in., 
and  the  angle  A  CB  =  52°  25'  are  measured.  Determine  the  angle  which 
AB  makes  with  AC.     Also  the  distance  AB.        Ans.   43°  15'.9,  159.27 

10.  To  determine  the  distance  between  two  ob- 
jects A  and  B,  a  base  line  CD  =  350  ft.  in  the  same 
plane  as  A  and  B  is  measured,  and  the  angles  B  CD  = 
40°  42',  ACB  =  S0^  30',  ADB  =  6r  12',  ADC  = 
32°  41',  are  observed.     Find  the  distance  AB. 

Ans.  273.4 

'  48.  Case  III.  Given  the  Three  Sides.  There  is  one  and 
only  one  solution,  provided  the  sum  of  any  two  of  the  given 
sides  is  greater  than  the  third  side. 


66 


PLANE  TRIGONOMETRY 


[VII,  §  49 


The  law  of  cosines,  applied  to  the  side  opposite  the  required 
angle,  will  always  give  a  solution ;  and  if  the  sides  are  small, 
or  if  only  one  angle  is  required,  it  is  often  the  best  method. 

^  Example  1.     Find  the  angles  of  the  triangle 

whose  sides  are  5,  7,  8. 
V5  By  the  law  of  cosines  : 

52  =  72  +  82 -2.  7.  8  cos  ^, 

A^ ^B  72  =  52  +  82  -  2  .  5  .  8  cos  J5, 

82  =  52  +  72  -  2  .  5  .  7  cos  0, 

cos  ^  =  Ji  =  .84615+    cos  ^  =  i,    cos  C  =  |  =  .14286- 

^  =  32°12'.3,   5  =  60°,    C  =  81°47'.2 
Check  :   A-\-  B+  C=  179°  59'.5 

Example  2.  The  sides  of  a  triangle  are  2431,  3124,  and  2314.  Find 
the  largest  angle. 

3124^  =  23l4^  +  243l^  -  2(2314)  (2431)  cos  a, 

2(2314) (2431) 
Call  the  numerator  x  and  the  denominator  y.    Then  the  solution  may  be 
carried  out  by  logarithms  as  f olfows  : 


^2314  = 

3.36436 
2 
6.72872 
5354500 
5909600 
11264100 
9759250 
1504850 
logx: 

logy: 

log  COS  a 

log  2431  = 

=  6.17750 
=  7.05117 
=  9.12633- 

=  3.38578 

2 

6.77156 

-10 

log  3124  =  3.49471 

2 

6.98942 

2314^  = 

2431^  = 

log  2  =  0.30103 
log2314  =  3.36436 

3124^  = 
x  = 

log2431  =  3.38578 
log?/ =  7.05117 

•.  a  =  82°  IS'.B 

49.  Logarithmic  Solution  of  Case  III.  To  compute  by  the 
aid  of  logarithms  the  three  angles  of  a  triangle  whose  sides 
are  known,  we  first  find  the  radius  of  the  inscribed  circle  by 
the  formula  of  §  43 : 


.^J(s-ct)(s-b)(s-c)^ 


VII,  §  49]      SOLUTION  OF  OBLIQUE  TRIANGLES  67 

and  then  compute  the  angles  by  the  formulas  of  §  42 : 


r 


tan  ^A  = ,   tan  ^B  = ,   tan  ^  C  =  - 

s  —  a  s  —  b  s  —  c 


Example.  Find  the  angles  of  the  triangle  whose  sides  are  2314,  2431, 
and  3124. 

The  work  may  be  arranged  as  follows 

a  =  2314  s  =  3934.5 

h  =  2431  s  -a  =  1620.5 

c  =  3124  s- 6  =  1503.5 

2s  =  7869  s-  c=    810.5 


Computation  of  log  r 

colog  s  =  6.40512  -  10 

log  (s- a)  =3.20965 

log  (s- 6)=  3.17710 

log(s  -c)=  2.90875 

log  7-2  =  5.70062 


2s  =  7869.0  (^Check) 
log  r  =  2.85031  log  r  =  2.85031 

log  (s-a)=  3.20965  log  (s  -  &)  =  3.17710 

log  tan  1  ^  =  9.64066  -  10  log  tan  i  ^  =  9.67321  -  10 

J  ^  =  23°  36' .8  i  J5  =  25°  13'.8 

log  r  =  2.85031 
log  (s-  c)  =  2.90875 
log  tan  A  C  =  9.94156  -  10 
iC  =  41°9'.4 
Then         ^  =  47°  13'.6,  5  =  50°27'.6,  0=82°  18'^ 

Check  :  i(A  +  B  +  G)  =90'' OO'.O 

EXERCISES  XVIL  — CASE  III 

1.  In  each  of  the  following  triangles,  the  three  sides  are  given.     Find 
the  smallest  angle. 

(a)  1,  2,  3.           -  Ans.   0° 

(6)  3,  5,  7.  Ans.   68°12'.8 

(c)  3,4,5.  Ans.   36°52'.2 

(d)  13,  14,  15.  Ans.  53°7'.8 

(e)  35,41,47.  Arts.  46°  15M 
(/)  4.7,  5.1,  5.8  Ans.  50°  35  .3 
(g)  48.3,  53.2,  62.7  -^^«-  48°24'.4 
(h)  1.9,3.4,4.9  ^^«-  16°25'.6 
(1)32.1,36.1,40.2  ^ris.  49°24'.0 
U)  5.29,  6.41,  7.02  ^^^s-   46°7'.0 

2.  Solve  each  of  the  following  triangles,  using  logarithms  : 

(a  )  a  =  22.2,      h  =  31.82,    c  =  40.64 

Ans.   32°54'.6,  51°8'.8,  95°  56'.6 
(6)  a  =27.53,    6  =  18.93,    c  =  30.14 

Ans.    63°31',  37°59M,  78°29'.9 


68 


PLANE  TRIGONOMETRY 


[VII,  §  50 


(c)  a  =523.8,    6  =  566.2,    c  =  938.4 

Ans.   29°17'.3,  31°  55' .5,  118°47'.3 

(d)  I    =3.171,  m  =  5.331,   n  =  5.101 

Ans.   35°18'.3,  76°  18' .6,  68°23'.l 

(e)  u  =40.04,    v  =  50.56,  to  =  70.12 

Ans.    34°7'.2,  45°5'.9,  100°46'.8 
(/)  p  =  38.2,      b  =  45.36,    d  =  26.54 

Ans.   57°  14' .7,  87°,  35°45'.2 
(g)  m  =  .126,     n  =  .3226,    c  =  .253 

Ans.   21°  11,  112°17'.8,  46°31'.2 
(A)    a  =.0506,    6  =  .1234,    c  =  .0936 

Ans.   21°  56',  114°2V.4,  43°42'.6 
(i)    w  =  167,       v  =  321,      to  =  231. 

^ns.   29°56'.4,  106°24'.3,  43°39'.3 
( j  )    u  =  196,1,    ?J  =  264.1,  w  =  135.4  Ans.    46°  3'.6,  29°  48'.8 

3.  Find  the  angle  subtended  by  a  rod  16.2  ft.  long  at  the  observer's 
eye,  which  is  11.9  ft.  from  one  end  and  17.6  ft.  from  the  other. 

Ans.    73°  44'. 

4.  To  determine  without  an  instrument  for  measuring  angles  the  angle 
between  two  lines  meeting  at  C,  the  distances  CA  =  500  ft.  and  CB  = 
700  ft.  are  measured  ;  AB  is  then  found  to  be  633  ft.     Find  Z  A  CB. 

Ans.   About  61°. 

5.  A  piece  of  land  is  bounded  by 
three  intersecting  streets,  on  which  the 
property  has  a  frontage  of  312  ft.,  472 
ft.,  and  511  ft.  respectively.  Find  the 
angles  at  which  these  streets  cross. 

Ans.  64°28'.4,  •77°40'.4,  37°51'.4 

6.  In  Fig.  57  AB  =  316.8  ft.,  BC  = 
^^^'^'^'                       226.4  ft.,  ^(7  =  431.6  ft.,    and  AD  = 

280.4  ft.     Find  BD.  Ans.   576.1 


50.  Case  IV.  The  Ambiguous  Case.  Here  we  have  given 
two  sides  and  the  angle  opposite  one  of  them ;  i.e.  an  angle,  a 
side  adjacent,  and  the  side  opposite. 

The  number  of  solutions  (two,  one,  or  none)  is  best  deter- 
mined by  the  geometrical  construction  of  the  triangle  from 
the  data. 

Construct  an  angle  AGQ,  equal  to  the  given  angle  which 
we  shall  at  first  suppose  to  be  acute ;  on  one  of  its  sides  lay 


VII,  §  50]     SOLUTION  OF  OBLIQUE  TRIANGLES 


69 


off  GA  equal  to  the  given  adjacent  side  and  drop  a  perpen- 
dicular AP,  to  the  other  side  OQ.  Then  with  A  as  center 
and  with  a  radius  equal  to  the  given  opposite  side  draw  an  arc. 
If,  as  in  Fig.  6^  (a),  this  arc  does  not  reach  GQ,  there  is 
no  solution;  if  it  is  tangent  to  GQ,  as  in  Fig.  58  (b),  there  is 
one  solution;  if  it  cuts  GQ  twice,  as  in  Fig.  58  (c),  there  are 
two  solutions;  if  it  cuts  GQ  once,  as  in  Fig.  58  (d),  there  is  one 
solution;  and  finally  if  the  given  angle  is  obtuse,  there  is  no 
solution  when  the  radius  of  the  arc  is  less  than  GA  and  one 
solution  when  it  is  greater. 


Fig.  58. 


The  results  may  be  collected  for  reference  as  follows :  Let 
G  =  the  given  angle,  (adj.)=  the  given  adjacent  side,  (opp.)  = 
the  given  opposite  side  ;  then 

I.  WJien  G  is  acute,  compute  p=:(adj.)  sin  G;  then  if 
(ppp.)<p  there  is  no  solution;  if  {opp.)  =  p,  one  solution; 
if  P  <  (pPP-)  <  i^^j')}  ^^0  solutions;  and  if  (opp.)  >  (adj.)  one 
solution. 

II.  JVhen  G  is  right  or  obtuse,  if  (opp.)  ^  (adj.),  there  is  no 
solution  but  if  (opp.)  >  (adj.),  one  solution. 

The  practical  method,  however,  in  the  case  of  any  given 
problem  is  to  construct  the  triangle  approximately  to  scale. 

Having  determined  the  number  of  solutions,  the  unknown 
parts  can  be  computed  by  the  law  of  sines. 


70 


PLANE  TRIGONOMETRY 


[VII,  §  50 


Fig.  59. 


Example  1.  Two  sides  are  12.56  and 
10.54  and  the  angle  opposite  the  latter  is 
64°  20'.     Solve  the  triangle. 

Construct  the  angle  G  =  64°  20'  and  lay- 
off GA  =  12.56  and  draw  AP.  A  glance 
at  the  tables  (p.  34)  shows  that -sin 
G  >  .9,  whence  p  >  .9  x  12.56  >  11. 
Therefore,  no  solution. 


Example  2.  In  the  triangle  ABC^  a  =  301.35,  c  =  352.11,  and  A  = 
33°  17'.     Determine  the  remaining  parts. 

Construct  angle  A  =  33°  17',  lay  off  AB  =  352.11,  and  draw  BP. 

Without  any  tables  whatever,  we  know  that  sin  33°  17'  <  .7  and  there- 
fore p  <  .7  X  360  <  260,  and  therefore  there  are  two  solutions. 


sin  G 


352.11 


sin  33°  17'     301.35 
log  sin  33°  17'  =  9.73940  -  10 
log  352. 11  =2.54668 
colog  301.35  =  7.52093  -  10 
log  sin  0  =  9.80701-10 
(7  =  39°53'.0 


Fig   60. 


There  are  two  angles  less  than  180°  having  a  given  sine ;  therefore  (7i  = 
39°  53'  and  C^  =  140°  7'. 

From  this  i)oint  on  we  have  to  solve  two  distinct  triangles,  viz.  :  ABCi 
and  ABC2.  Call  AGi,  61,  and  AC2,  62  ?  angle  ABC\^  Bi  and  angle 
ABG2 ,  ^2.     Then  B^  =  106°  50'  and  B^  =  6°  36'. 


61     ^  sin  106°  50' 
301.36      sin  33°  17'  * 
log301.35  =  2.47907 
log  cos  16°  50'  =  9.98098  -  10 
colog  sin  33°  17'  =  0.26060 
log  61  =  2.72065 
61  =  525.59 


sin  6°  36' 


301.35     sin  33°  17' 
log  301.35=  2.47907 
log  sin  6°  36'  =  9.06046  -  10 
colog  sin  33°  17'  ±=  0.26060 
log  62  =  1.80013 
&2  =  63.114 


Example  3. 


Two  sides  of  a  triangle  are  5  and  7  and  the  angle  oppo- 
site the  latter  is  120°.     Solve  the  triangle. 

Construct  the  angle  G  =  120°,  lay  off 
GA  =  5.  It  is  at  once  obvious  that  a 
circle  center  at  A,  of  radius  7,  will  cut 
G  Q  once  and  only  once,  at  B. 

Let  the  student  complete  the  solution, 
finding  by  the  law  of  sines,  angle  B  and 
the  side  GB.  Ans.    38°  12'.8,  3. 


1. 

a  =  17.16, 

b  =  14.15, 

2. 

a  =  54, 

6  =  48.6, 

3. 

u  =  971, 

V  =  1191, 

4. 

I  =  281, 

m  =  152, 

6. 

b  =  13.12, 

c  =  7.22, 

6. 

P=  48, 

q  =  36.1, 

7. 

m  =  10.08, 

n  =  5.82, 

8. 

i  =  93.99, 

8  =  91.97, 

9. 

a  =  309, 

b  =  360, 

10. 

k  =  91.06, 

m  =  77.04, 

11. 

One  diagor 

lal  of  a  para 

VII,  §  50]     SOLUTION  OF  OBLIQUE  TRIANGLES  71 

EXERCISES   XVIII.  — CASE  IV 

Solve  each  of  the  following  triangles,  using  logarithms ;  if  two  solu- 
tions exist,  obtain  both  of  them. 

5  =  42^.  Ans.   83°45'.7,  21.022 

A  =  Sr  14'.       Ans.   120°  56'.9,  89.314 
Z7=51°15'.  Ans.   55°  41'. 8,  1028.5 

L  =  103°.  A71S.   45°  11'.6,  204.61 

5  =  39°  54'.  Ans.  20°  40'. 2,  17.814 

Q  =  45°50'.  Ans.   61°  39'. 5,  44.293 

if  =21°  31'.        Ans.    146°  15^4,  15.264 
r=120°35'.  Ans.  2°  1'.3,  3.85 

^  =  21°14'.4      Ans.   133°47'.7,  615.67 
Ji  =  51°9'.l  Ans.   87°37'.9,  116.82 

ilelogram  is  68  ft.  long  and  makes  an  angle 
of  30°  20'  with  the  other  diagonal ;  one  side  is  22  ft.  long.  Find  the 
length  of  the  other  side.  Ans.   48.107  or  74.450 

12.  In  a  certain  town  the  streets  intersect  at  an  angle  of  82°  14'.  It 
is  desired  to  know  the  distance  between  two  objects,  A  and  5,  which  lie 
on  a  line  parallel  to  one  set  of  streets  and  which  are 
separated  by  a  large  building.  A  line  AG  =  200  ft. 
is  measured  along  a  side  line  parallel  to  the  other  set 
of  streets,  and  CB  =  222  ft.  is  then  measured.  De- 
termine AB.  Ans.    127.09 

13.  The  pilot  of  a  ship  S  sees  a  lighthouse  H  on 
the  shore  ;  by  measm-ing  the  angle  of  elevation  of  the 
top  of  the  lighthouse,  and  knowing  its  height,  he  de- 
termines that  it  is  8950  ft.  from  his  ship.     At  the  ship  ^^^'  "^* 

an  angle  of  2°  40'  is  subtended  by  a  line  connecting  the  lighthouse  with  a 
light  L  on  the  shore  known  to  be  575  ft.  from  the  lighthouse.  Find  the 
angle  SLH  and  thus  determine  exactly  the  position  of  the  sliip  with 
reference  to  the  shore.  Practically,  how  may  he  tell  which  of  the  two 
possible  solutions  is  actually  correct  ?  Ans.    46°  24',  133°  36'. 

14.  Suppose  a,  6,  and  A  are  given ;  let  x  represent  the  third  side. 
Apply  the  law  of  cosines  to  side  a  and  determine  under  what  conditions 
the  resulting  equation  in  x  will  have  (1)  no  real  root,  (2)  one  positive 
real  root,  (3)  two  positive  real  roots.  Consider  separately  the  two  cases 
when  A  is  acute  and  when  A  is  obtuse  and  compare  results  with  the 
statements  of  §  50,  p.  68. 


CHAPTER   VIII 
AREAS  —APPLICATIONS  —  PROBLEMS 

51.  Areas  of  Triangles.  It  is  shown  in  plane  geometry 
that  the  area  A,"^  of  a  triangle  is  equal  to  one-half  the  product 
of  any  side  and  the  altitude  from  the  opposite  vertex. 

(1)  The  area  of  a  triangle  is  equal  to  one-half  the  product  of 
the  base  and  altitude. 

52.  Area  from  Two  Sides  and  the  Included  Angle.    If  we 

have  two  sides  and  the  included  angle,  a,  b,  and  C,  and  drop 


a  perpendicular  upon  one  of  the  given  sides,  as  p  upon  b, 
then  p=  asin  C  and  by  (1)       A  =  ^ 6 (a  sin  C) ;      whence 

(2)   The  area  of  a  triangle  is  equal  to  one-half  the  product  of 
any  two  sides  into  the  sine  of  their  included  angle, 

53.  Area  from  Three   Sides.     If 

the  three  sides  are  given,  draw  lines 
from  the  vertices  to  the  center  of 
the  inscribed  circle  dividing  the  tri- 
angle into  three  triangles  having  a 
common  altitude,  r.     By  (12),  §  43, 


.  ^    l(s-a)(8-&)(g-c)  _ 


Fig.  65. 


♦The  area  is  denoted  by  the  boldface  type  A  in  distinction  from  the  angle  A. 

72 


VIII,  §  54] 


AREAS 


73 


The  sum  of  the  bases  of  the  three  triangles  is  a-|-6  +  c  =  2s. 
Therefore  their  combined  area  is,  by  (1), 

(3)  A  =  rs  =  V<s  -  a)(s  ~-b){s-  c). 

Hence  we  have  the  rule : 

(3)  Add  the  three  sides  and  take  half  the  sum;  from  the  half 
sum  subtract  the  three  sides  severally;  take  the  product  of  the  half 
sum  and  the  three  remainders  and  extract  the  square  root. 

54.  Illustrative  Examples.  The  area  is  most  conveniently 
found  in  other  cases  by  solving  the  triangle  sufficiently  to 
secure  the  data  required  by  one  of  the  three  rules  given  above, 
all  of  which  are  adapted  to  logarithmic  computation. 

Example  1.  One  side  of  a 
triangle  is  50,  the  angle  oppo- 
site is  10°  12',  and  another  angle 
is  46°  36'.     Find  the  area. 

The  third  angle.  A,  Fig.  66, 
is  then  123°  12'.  If  we  knew 
a  or  6,  we  should  know  two 
sides  and  the  included  angle. 
By  the  law  of  sines, 

a  ^  sin  123°  12' 
50 


sin  10°  12' 
By  rule  (2)  the  area, 

A  =  i  .  50  .  a  .  sin  46°  36', 


log  50  =  1.69897 
log  cos  33°  12'  =  9.92260  -  10 
colog  sin  10°  12'  =  0.75182 
log  a  =  2.37339 
log25  =  1.39794 
log  sin  46°  36'  =  9.86128  -  10 
log  A  =  3.63261 
A  =  4291.5 
Example  2.     Two  sides  of  a  triangle  are  35  and  50  and  the  angle  oppo- 
site the  latter  is  28°  30'.     Find  the  area. 

On  constructing  the  triangle.  Fig.  67,  it  is  evident  that  there  is  only 
one  solution  and  B  is  acute.    By  the  law  of  sines, 

sin^     _35^ 

~      *  log  sin  28°  30'  =  9.67866  -  10 

log35  =  1.54407 
colog  50  =  8.30103  -  10 
log  sin  5  =  9.52376 -10 

5  =  19°30'.7 

^       ^^  whence  A  =  131°  59'.3 

Fig.  67.  . 


sin  28°  30'      50 


74  PLANE  TRIGONOMETRY  [VIII,  §  54 

We  now  know  two  sides  and  the  included  angle  and 
A  =  i  •  50 .  35  .  sin  131°  59'.3  log  25  =  1.39794 

log  35  =  1.54407 
log  cos  41°  59'.3  =  9.87115-10 
A  =  650.37  log  A  =  2.81316 

Example  3.     The  sides  of  a  triangle  are  13,  37,  and  40.     Find  the  area. 
Using  (3),  we  have. 


A  =  \/s(8- 

a)(s-6)(s- 

c), 

and  the  computation  may  be  made  as  follows  : 

s 

=  45 

log  = 

=  1.65321 

a  =13 

s  —  a 

=  32 

log  = 

=  1.50515 

6  =  37 

s-b 

=    8 

log  = 

--  0.90309 

c  =  40 

s  —  c 

=    5 

log  = 

:  0.69897 

2s  =  90 

Check 

=  90 

logA  = 
A  = 

:  2.38021 

:2400 

EXERCISES  XIX.  — AREAS 

Find  the  area  of  the  following  triangles : 

1. 

a  =  829, 

6  =  592, 

C  =  62°. 

Ans.   216,661. 

2. 

a  =  713, 

6  =  987, 

c  =  1255. 

Am.   351,105. 

3. 

B  =  25°, 

C  =  68°, 

6  =  392. 

Ans.    168,331. 

4. 

p  =  231, 

q  =  195, 

P  =  47°. 

Ans.   22,440. 

5. 

u  =  S, 

V  =  6, 

Tr=60°. 

Ans.    17.32 

6. 

k  =  72.3, 

^=52°  35, 

M  =  63°  17'. 

Ans.   2648.7 

7. 

I  =  .582, 

m  =  .601, 

n  =  .427 

Ans.    0.11765 

8. 

6=21.5, 

c  =  30.456, 

D  =  41°  22'. 

Ans.   216.37 

9. 

w  =  41. 

V  =  401, 

w  =  408. 

Ans.    8160. 

10. 

p  =  62.4, 

q  =  20.5, 

r  =  44.5 

Ans.   262.08 

11. 

A  =  60°, 

6  =  30, 

a  =  70. 

Ans.    1039.23 

12. 

a  =  78.35 

,  5  =  34°  22', 

C  =  66°  11'. 

Ans.    1613.3 

13. 

p  =  26.6, 

q  =  35.2, 

E  =  73°. 

Ans.   447.7 

U. 

Find  the 

area  of  a  triangular  field  having  one  of  its  sides  15  rods 

in  length,  and  the  two  adjacent  angles,  respectively,  70^ 

'and  69°  40'. 

Ans.    153.16 

15. 

The  area  ( 

3f  a  triangular  plat  of  ground 

is  one 

acre.     Two  of  its 

sides  are  127  yd.  and  150  yd.,  respectively.     Find  the  angle  between  them. 

Ans.   30°  32'. 4 
16.   The  length  of  the  bisector  of  one  of  the  acute  angles  of  an  isosceles 
right  triangle  is  4.    Find  the  area.  Ans.  4. 


VIII,  §  55]  AREAS  75 

55.   Composition  and  Resolution  of  Forces  and  Velocities. 

We  saw  in  §  27  that  forces  and  velocities  may  be  represented 
graphically  by  straight  line  segments.  The  length  of  such  a 
segment  represents  the  magnitude  of  the  force  or  velocity, 
and  its  direction  the  direction  of  the  force  or  velocity. 

To  find  the  effect  of  two  simultaneous  velocities,  let  us  sup- 
pose that  a  body  moves  along  a  straight  track  with  a  velocity 
of  4  units  per  second  and  that  each  point  of  the  track  moves 
with  a  velocity  of  3  units  per  second  along  a  line  making  an 
angle  of  60°  with  the  track.  What  is  the  position  of  the  body 
at  the  end  of  1  second  ?  To  answer  this  question  draw  a  seg- 
ment 4  units  long  to  represent  the  magnitude  and  direction 
of  the  velocity  of  the  body  along  the  track,  and  from  the  ends 
of  this  segment  draw  segments  AO,  BD,  each  3  units  in  length 
and  making  an  angle  of  60°  with  AB  to 
represent  the  magnitude  and  direction 
of  the  velocity  of  the  ends  of  the  track. 
The  track  will  then  take  the  position 

CD  at  the  end  of  1  second.     But  since  ^ 

Fig.  68. 

the  body  moves  along  the  track  at  the 

rate  of  4  units  per  second,  it  will  reach  the  point  D  at  the  end 
of  1  second.  That  is,  it  will  reach  the  same  point  as  if  it  had 
moved  along  the  diagonal  AD  with  a  speed  represented  by  the 
length  of  the  diagonal.  The  velocity  represented  by  AD  is 
called  the  resultant  of  the  velocities  represented  by  AB  and 
AC,  AB  and  AC  are  called  components.  The  length  of  AD 
can  be  computed  by  solving  the  triangle  ABD. 

The  resultant  of  any  two  velocities  may  be  found  by  draw- 
ing from  a  common  point  A,  segments  AB,  AC  to  represent  the 
given  velocities  in  magnitude  and  direction  and  then  complet- 
ing the  parallelogram  ABCD.  The  diagonal  AD  represents 
the  resultant.     This  fact  is  often  called  the  parallelogram  law. 

The  resultant  of  two  forces  is  found  by  a  similar  construc- 
tion.    This  diagram  is  known  as  the  parallelogram  of  forces. 


76 


PLANE  TRIGONOMETRY 


[VIII,  §  56 


66.  Illustrative  Examples.  Example  l.  The  angle  between  the 
directions  of  two  forces  of  19  lb.  and  26  lb.  is  64°.  Find  the  magnitude 
and  direction  of  their  resultant. 

The  forces  may  be  represented  by  segments  19  units  long  and  26  units 
long,  respectively,  and  making  the  angle  of  54°  with  each  other.     If  the 

Q parallelogram  is  completed  which  has  these  seg- 

^p  /  ~^^^^     ments  for  two  of  its  intersecting  sides,  the  diag- 

onal extending  from  their  intersection  to  the 
opposite  corner  will  represent  the  resultant  both 
in  magnitude  and  in  direction.  This  diagonal  is  a 
side  of  a  triangle  having  two  sides  equal  to  19 
and  26,  respectively,  with  an  included  angle  of  126°  (the  supplement 
of  54°).     Hence  we  can  find  the  magnitude  and  direction  of  the  resultant. 

Example  2.  Two  forces  of  51  lb.  and  73  lb.  have  a  resultant  of  80  lb. 
Find  the  angle  between  them. 

In  this  case,  in  the  parallelogram  of  forces,  the  diagonal  and  two  inter- 
secting sides  are  known  ;  the  angle  opposite  the  diagonal  is  determined 
by  Case  III.     The  required  angle  is  the  supplement  of  this  one. 

Example  3.     A  weight 

of  100  lb.  is  supported  by 

two  cords  AW^  3  ft.  long, 

and   5Tf,  5  ft.  long,  at- 
tached   to    a    horizontal 

beam  at  A  and  J5,  7  ft. 

apart.     Find  the  tensions, 

sin^TFand  t\\\  BW. 

Since  the  weight  acts  vertically,  we  need  the  angles  a  and  ^  which 

AW  and  BW  make  with  the  vertical  WP.     Solving  the  triangle  ABW, 

we    find  ^  =  38' 12'. 8,    2?  =  21°  47'. 2,   whence    a  =  51°47'.2    and  /3  = 

68°12'.8 

To  construct  Fig.  71,  draw  AG 
making  the  angle  ct  =  51°  47'.2  with 
the  vertical  and  similarly  draw  AD 
making  ^  =  68°  12'.8.  Take  AV  = 
100  on  some  convenient  scale  and 
draw  VE  parallel  to  AD  and  VF 
parallel  to  AG.  Then  AE  repre- 
sents s  and  AF,  t;  because  a  force 
of  100  lb.  acting  upward  at  A  must 
be  the  resultant  of  s  and  t  since  the 

point  A  is  at  rest.     In  the  trmngles  AVE  and  AVE  we  have  enough 

data  to  find  s  =  104.8  and  t  =  90.7 


Fig.  70. 


VIII,  §  56] 


FORCES  AND  VELOCITIES 


77 


EXERCISES   XX.  — VECTORS 


40.22,  22°28M  with  AB. 
Ans.   10P51'.4 


75  Lbs. 


100  Lbs. 


1.  Solve  Example  1,  above.  Ans. 

2.  Complete  the  solution  of  Example  2. 

3.  Compute  cc,  /3,  s,  and  t  of  Example  3. 

4.  Check  the  answers  to  Example  3  by  finding,  (a)  the  sum  of  the 
vertical  components  of  s  and  t  ;    (6)  their  horizontal  components. 

5.  Three  forces  of  13  lb.,  22  lb.,  and  28  lb.,  respectively,  are  in  equi- 
librium.    Determine  the   angles  which   they  make  with   one  another. 

[Hint.     Study  Example  2.]  Ans.   76°46'.6,   130^6^4,   153°7'.8 

6.  Find  the  resultant  of  two  forces  of  30  lb.  and  40  lb.  acting  at  an 
angle  of  60°  with  each  other.  Ans.   60.83 

7.  A  ball  rolls  along  the  diagonal  of  the  floor  of  a  car  from  the  back 
to  the  front  with  a  speed  of  30  ft.  per  second.  The  car  is  moving  forward 
with  a  speed  of  40  ft.  per  second.  Find  the  actual  speed  of  the  ball  if 
the  car  is  7  ft.  wide  and  30  ft.  long. 

Ans.   69.55 

8.  Two  forces  are  acting  on  a 
block  resting  on  the  ground  as 
shown  in  the  figure.  What  hori- 
zontal force  could  replace  them  ? 

Ans.  139.85 

9.  A  point  is  kept  at  rest  by 
forces  of  6,  8, 11  lb.  Find  the  angle 
between  each  pair.  Ans.   77°21'.9,  147°  60' .6,  134°  47^.6 

10.  A  boat  is  rowed  across  a  river  at  the  rate  of  3.5  mi.  per  hour  ;  the 
river  flows  at  the  rate  of  4.8  mi.  per  hour.  Find  the  speed  of  the  boat 
and  the  direction  of  its  motion.  Ans.   5.94,  36°  6'  with  shore. 

11.  A  ship  is  sailing  10  mi.  per  hour  and  a  sailor  climbs  the  mast 
200  ft.  high  in  30  sec.  Find  his  speed  relative  to  the  earth,  and  the  direc- 
tion of  his  motion.     Ans.  966.7  ft.  per  min.,  24°  26^6  with  the  vertical. 

12.  A  train  is  going  15  mi.  per  hour  northward  ;  a  man  crosses  the 
car  eastward  12  ft.  per  second.  Find  his  speed  relative  to  the  ground, 
and  his  direction.  Ans.   25.06  N.,  28°  36'.6  E. 

13.  A  ball  rolling  along  the  floor  10  ft.  per  second  is  struck  so  that  its 
speed  is  increased  2  ft.  per  second,  and  the  direction  of  motion  is  changed 
45°.     What  speed  and  direction  of  motion  is  due  to  the  stroke  alone  ? 

.     Ans.   8.6  ft.  per  second,  80°  30'. 

14.  A  river  flows  4  mi.  per  hour,  and  a  motor  boat  goes  6  mi.  per  hour. 
In  what  direction  must  the  boat  be  pointed  to  go  straight  across  the  river, 
and  what  will  be  its  speed  ?  Ans.  63°  36'. 7,  8.06  mi.  per  hom\ 


Fig.  72. 


78 


PLANE  TRIGONOMETRY 


[VIII,  §  56 


15.  An  oarsman  rows  his  boat  due  north  5  miles  an  hour.  There  is  a 
breeze  of  12  miles  an  hour  from  the  southeast.  Determine  the  resulting 
speed  and  direction  of  the  boat  if  the  resistance  of  the  water  damps  the 


effect  of  the  wind  one-third. 


R<- 


Ans.  6.03  mi.  per  hour,  N.  27°  58'  W. 

16.  A  weight  of  400  lb.  is  drawn  along 
the  ground  by  a  force  of  600  lb.  attached 
as  shown  in  Fig.  73.  What  pressure 
does  it  exert  on  the  ground  ?  If  the 
resisting  force  B  (due  to  friction)  ia  1% 
of  the  pressure  on  the  ground,  what  re- 


FiG.  73. 
sultant  force  is  effective  in  moving  the  weight  forward  ? 

Ans,    1501b.,  408  1b. 


EXERCISES  XXL  — MISCELLANEOUS  PROBLEMS 


1.   Solve  t 

he  following  tr 

iangles  : 

(a) 

a  =  T0.34, 

1^  =  5°  7'.6, 

0  =  19°  49'. 

W 

a  =  36.423, 

b  =  14.678, 

C  =  68°  14'. 

(O 

I  =  14.236, 

m  =  13.761, 

i\r=45°ll'. 

(d) 

a  =  734.34, 

B  =  108°  6', 

0  =  61°  7'. 

(O 

u  =  32.19, 

V  =  69.182, 

Z7=:69°17'. 

(/) 

a  = .75632, 

b  =  .62761, 

0=84°  48'. 

^9) 

c  ==  454.72, 

j=iriv, 

0=57°  37'. 

(h) 

a  =  474.17, 

b  =  1008.8, 

c  =  940.25 

(O 

a  =  100.37, 

c  =  95.376, 

B  =  100°  58'. 

U) 

d  =  391.68, 

D  =  25°  36', 

B  =  68°  13'. 

W 

a  =  622.02, 

b  =  293.22, 

A  =  100°. 

(0 

u  =  375.64, 

V  =  438.79, 

w  =  133.94 

(m) 

a  =  .010231, 

C  rz:  .0047233, 

^  =  44°  58'. 

(n) 

a  =  476.53, 

P=40°17', 

A  =  39°  14'. 

(0) 

b  =  94.961, 

a  =  88.234, 

c  =  12°. 

(P) 

b  = .43124, 

a  =  .53467, 

JL  =  99°  69'. 

(a)  2.1909,  8.3119 

(c)  69°44'.6,  65°4'.4,  10.764 

(e)  No  solution. 

(g)  104.43,  502.02 

(i)  40°43'.3,  38°18'.7,  151.04 

(k)  27°39'.6,  52°20'.4,  500.01 

(m)  115°  59'. 5,  19°  2'.6,  0.013013 

(o)  64°44'.5,  103°15'.5,  20.284 


Answers  to  the  preceding  exercises. 

(6)  88°11'.2,  23°34'.8,  33.844 

(d)  3730.7,  3436.7 

(/)  53°25'.2,  41°46'.8,  0.93795 

(h)  27°52'.6,  84°7'.7,  67°59'.7 

(j)  904.48,  841.76 

(I)  53°49'.8,  109°26'.4,  16°43'.8 

(n)  487.13,  740.85 

(p)  52° 36'. 5,  27°25'.5,  0.25005 


VIII,  §  56]         MISCELLANEOUS  PROBLEMS  79 

2.  A  pole  17  ft.  high  has  a  mark  8  ft.  4  in.  from  the  ground.  Find 
the  angle  subtended  by  each  part  at  a  point  20  in.  from  the  ground  and 
53  ft.  4  in.  from  the  pole.  Ans.   8°  64'.9 

3.  The  diagonals  of  a  parallelogram  are  22  ft.  and  31  ft.,  and  the 
angle  between  them  is  51°  12^     Determine  the  sides  of  the  parallelogram. 

Ans.    23.977,  12.148 

4.  A  biplane  is  observed  from  the  ground  and  from  an  upper  window 
of  a  building  60  ft.  directly  above.  The  angles  of  elevation  are  found  to 
be  10°  42'  and  9°  58^     Find  the  distance  from  each  point  to  the  airship. 

Ans.   4606.4,  4617.2 

5.  Two  sides  of  a  triangle  are  63  and  81,  and  the  included  angle  is 
54°.     Find  the  length  of  the  bisector  of  the  largest  angle.    Ans.   61.015 

6.  The  sides  of  a  triangle  are  22,  35,  44.  Find  the  length  of  the 
median  to  the  longest  side.  Ans.    20.5 

7.  Two  sides  of  a  triangle  are  7.2  and  8.1  and  the  angle  opposite  the 
latter  is  32°  41'.  Find  the  radius  of  the  circumscribed  circle.    Ans.   15. 

8.  The  three  sides  of  a  triangle  are  26,  28,  and  30.  Find  the  radius 
of  the  inscribed  circle.  -^^^^  Ans.    8. 

9.  The  angles  of  a  triangle  are  to  each  other  as  1  : 2  : 3  ;  the  altitude 
upon  the  longest  side  is  45.     Find  the  sides.         Ans.   90,  51.96,  103.92 

10.  The  sides  of  a  triangle  are  to  each  other  as  2:3:4.  Find  the 
angles.  Ans.   28°  57'. 3,  46°  34',  104°28'.7 

11.  To  determine  the  distance  between  two  objects  A  and  B  that  have 
a  barrier  between  them,  a  distance  AC  =  200  ft.  is  measured  to  a  point  (7, 
from  which  both  objects  are  visible.  The  distance  BC  =  321  ft.  and  the 
angle  ACB  =  68°  41'.     Find  the  distance  AB.  Ans.   310.43 

12.  To  find  the  distance  between  two  objects  A  and  B  situated  on  op- 
posite sides  of  a  lake,  the  distance  AC  =  250  ft.  and  the  angles  CAB  = 
44°  13',  ACB  =  51°  9',  are  measured.     Find  AB.  Ans.    195.65 

13.  An  object  B  is  wholly  inaccessible  and  is  invisible  from  a  certain 
point  A.  To  find  the  distance  AB,  two  points  C  and  D,  from  which  B 
can  be  seen,  are  selected  on  a  line  through 
A.  If  CD  =  243  ft.,  CA  =  102  ft.,  Z  DCB 
=  68°  56',  Z  CDB  =  48°  22',  find  AB. 

Ans.    192.9 

14.  It  is  desired  to  know  the  height  of 
an  object  AB.     A  line   CI>  =  260  ft.,  in  a 

^horizontal  plane  with  the  base  A  of  the 
object,  is  measured,  also  the  angle  of  eleva- 
tion ACB  =  13°  22',  and  the  angles  DCA  = 
35°  37'  and  CDA  =  64°  28'.     Determine  the  height  AB.        Ans.   64.44 


80 


PLANE  TRIGONOMETRY 


[VIII,  §  56 


15.  A  tall  building  stands  at  the  foot  of  a  hill.  From  a  point  on  the 
side  of  the  hill  the  angle  of  depression  of  the  base  of  the  building  is  ob- 
served to  be  14°  36^  and  the  angle  of  elevation  of  the  top  is  21°  43'.  A 
level  line  from  the  instrument  meets  the  building  19  ft.  7  in.  above  the 
base.    Find  the  height  of  the  building.  Ans.   49.62 

16.  A  balloon  is  observed,  at  the  moment  it  passes  over  a  level  road, 
from  two  points  in  the  road  an  eighth  of  a  mile  apart.  The  angles  of  ele- 
vation from  the  two  points  are  33°  11'  and  42°  6'.  Find  the  distances  of 
the  balloon  from  the  two  observers.  Arts.   374.31,  427.26 

17.  In  surveying,  it  is  sometimes  desired  to 
extend  such  a  line  as  AB  in  the  figure  beyond 
an  obstacle.  If  at  J5  a  right  turn  of  58°, 
BE  =  126  ft.,  and  at  J5;  a  left  turn  of  110°  are 
laid  off,  compute  EC^  and  the  angle  (right 
turn)  at  C.  Ans.   135.6,  52°. 

18.  To  find  the  distance  PQ  in  Fig.  76,  a 
base  line  AB  is  measured  =  518  ft.  At  A 
the  angles  PAQ  =  43°  18'  and  QAB  =  48°  32' 
are  measured  and  checked  by  measuring 
P^B  =  91°50',  and  at  5,  ^J5P  =  38°  43', 
P5Q  =  41°28',  ABQ  =  SO""  11'.  Find  PQ 
by  two  methods.  Ans.  451.39 

19.  Find  the  distance  A  (7,  Fig.  77,  through 
a  thicket,  having  measured  AB  =  20.71  rods, 

BG=  18.87  rods,   angle 


Fig.  76. 


ABC  =  6^° 


12'. 
Ans. 


18.40 


Fig.  77. 


20.   From  two  points  A  and  B,  300  ft.  apart  on 

the  deck  of  a  ship,  a  second  ship,  S,  is  observed. 

The  angles  ABS  =  85°  18',  and  BAS  =  83°  47'  are 

measured.     What  is  the  distance  between  the  ships  ? 

Ans.   2496,  2502,  av.  2499. 


21.  How  far  to  the  side  of  a  target  1300  ft.  away  should  a  gunner  aim 
from  a  ship  going  15  mi.  per  hour,  if  the  speed  of  the  bullet  is  2000  ft. 
per  second  and  he  fires  when  he  is  directly  opposite  ?  Ans.    14.3 

22.  From  a  railway  train  going  50  mi.  per  hour  a  bullet  is  fired  1000  ft. 
per  second  at  an  angle  of  75°  28'. 3  with  the  track  ahead.  Find  its  speed 
and  direction.  Ans.   71°  29'.2,  1020.9  ft.  per  second. 

23.  A  man  in  a  railway  car  going  45  mi.  per  hour  observes  the  rain- 
drops falhng  at  an  angle  of  30°  with  the  vertical.  Assuming  that  the 
raindrops  are  actually  falling  vertically,  find  their  speed.        Ans.   77.9 


VIII,  §  56]  MISCELLANEOUS  PROBLEMS  81 

24.  The  resultant  of  two  forces  is  10  lb. ;  one  of  the  forces  is  8  lb.  and 
makes  an  angle  of  36°  with  the  resultant.  Find  the  magnitude  of  the 
other  force.  Ans.  6.88 

25.  A  horse  pulls  a  canal  boat  by  a  rope  which  makes  an  angle  of 
25°  36'  with  the  tow  path.  What  size  of  engine  would  propel  the  boat  at 
the  same  speed  ?     (Assume  that  the  horse  is  doing  one  '*  horse  power.") 

Ans.   0.9+ 

26.  A  man  climbs  a  hill  inclined  (on  the  average)  32°  with  the  hori- 
zontal. His  pocket  barometer  shows  that  at  the  end  of  2 J  hr.  he  has 
increased  his  elevation  2760  ft.     Find  his  average  speed  up  the  slope. 

Ans.   2076.8 

27.  The  sides  of  a  triangular  field  are  82.7  rods,  91.4  rods,  and  104.3 
rods.     Determine  the  area  of  the  field  and  the  angles  between  the  sides. 

Ans.    226.39  A. ,  49°  27^4,  67°  7^6,  73°  26^ 

28.  Find  the  area  of  a  triangular  piece  of  ground,  having  two  angles, 
respectively,  73°  10'  and  90°  60',  and  the  side  opposite  the  latter  160.6  rods. 

Ans.    18.7  A. 

29.  Find  the  areas  of  triangles  which  have  the  following  given  parts j 
(a)  a  =  116.082,  6  =  100,  C=118°16'.7 

(6)6  =  100,  A  =  76°  38'.2,  C  =  40°  6'. 

(c)  w  =  31.326,  «  =  13°67',  i7=63°ll',3 

(d)  a  =  408,  6  =  41,  c  =  401. 

(e)  a  =.9,  6  =  1.2,  c  =  1.6 

Ans.    (a)  6112.1     (6)  3606.8     (c)  136.13     (d)  8160     (e)  .54 

30.  Three  circles  whose  radii  are  2,  3,  10,  respectively,  are  tangent 
externally.     Find  the  area  of  the  triangle  formed  by  joining  their  centers. 

Ans.   30. 

31.  Prove  that  the  area  of  the  triangle  formed  by  joining  the  centers 
of  any  three  circles  which  are  tangent  externally  is  a  mean  proportional 
between  the  sum  and  the  product  of  their  radii.     See  §  63. 

32.  Prove  that  one-half  the  product  of  the  three  sides  of  any  triangle 
is  equal  to  the  product  of  its  area  into  the  diameter  of  its  circimiscribed 
circle.     See  §§40  and  62. 

33.  Prove  that  the  area  of  any  triangle  is  equal  to  the  product  of  the 
radii  of  its  inscribed  and  circumscribed  circles  into  the  sum  of  the  sines 
of  its  angles.     See  §§40  and  63. 


PART  m.     THE   GENERAL  ANGLE 

CHAPTER   IX 
DIRECTED   ANGLES  — RADIAN   MEASURE 

57.  Directed  Lines  and  Segments.  As  explained  in  ele- 
mentary algebra,  it  is  often  convenient  to  select  one  direction 
on  a  straight  line  as  the  positive  direction;  the  other  is  then 
called  the  negative  direction.  Thus,  if  two  forces  act  along  the 
same  line,  but  in  opposite  directions,  it  is  convenient  to  call 
one  positive  and  the  other  negative. 

Two  segments  are  said  to  have  the  same  sense  if  they  lie  on 
the  same  line  or  on  parallel  lines,  and  if  both  are  positive  or 
both  are  negative.  Two  segments  are  said  to  be  of  opposite 
sense  if  they  lie  on  the  same  line  or  on  parallel  lines,  and  if 


4- 


Q    Q    Q  one  is  positive  and  the  other  is  negative. 

^"■^"^I;       "       Thus,    in    Fig.     78,    AB  =  EF,    while 

^ +--+— ^ ^   ^(7  =  -G^^.  and  CB=   -  FQ, 

The  numerical  measure  of  a  directed  seg- 
ment is  the  number  of  units  in  its  length  with  the  sign  +  or 
— ,  according  as  the  segment  is  positive  or  negative. 

58.  Rotation.  Directed  Angles.  In  describing  rotation,  it 
is  convenient  to  regard  angles  as  positive  or  negative  in  a 
manner  analogous  to  that  explained  in  §  57  for  line-segments. 

An  angle  is  thought  of  as  generated  by  the  rotation  of  one 
of  its  sides  about  the  vertex  as  center;  its  first  position  is 
called  the  initial  side,  the  final  position  is  called  the  terminal 
side.  An  angle  generated  by  a  rotation  opposite  to  the  motion 
of  the  hands  of  a  clock  (counterclockwise) j  is  said  to  he  positive; 

82 


IX,  §  59]         DIRECTED  LINES  AND  ANGLES 


83 


an   angle   generated   by   a  clockwise   rotation,   is   said   to  be 
negative.* 

Angles  may  be  of  any  magnitude,  positive  or  negative. 
Thus,  in  Fig.  79,  a,  /3,  8  are  positive  angles ;  y  is  negative ; 
fi  is  greater  than  a  straight  angle  ;  and 
8  is  greater  than  360°,  or  a  complete 
revolution.  In  rotating  parts  of  ma- 
chinery, such  angles  have  a  very  vivid 
meaning.  Thus,  a  wheel  which  rotates 
370°  per  second  has  a  very  different 
speed  from  that  of  a  wheel  which  rotates 
10**  per  second.  yig.  79. 

59.  Placing  Angles  on  Rectangular  Axes.  To  place  any 
given  angle  on  a  pair  of  rectangular  axes  in  the  plane  of  the 
angle,  put  the  vertex  at  the  origin  and  the 
initial  side  on  the  a>-axis  extending  to  the 
right ;  the  terminal  side  will  then  fall  in  one 
of  the  four  quadrants  (or,  if  the  angle  is  a 
multiple  of  a  right  angle,  on  one  of  the  axes). 
If  the  terminal  side  falls  in  the  first  quad- 
rant, the  angle  is  said  to  be  an  angle  in  the 
first  quadrant,  etc.  In  Fig.  80,  a  is  a  positive 
angle  in  the  first  quadrant,  ^  is  a  negative  angle  in  the  fourth 
quadrant,  8  is  a  positive  angle  in  the  fourth  quadrant. 


Quad. 
II 

f 

/Quad. 

(\ 

Act  \ 

\   0 

Quad.V. 

A    Q^ad. 

A" 

Fig.  80. 


EXERCISES  XXII.  — DIRECTED  LINES  AND  ANGLES 

1.  What  angle  will  the  minute  hand  of  a  clock  generate  in  2  hr.  24 
min.  10  sec.  ? 

2.  A  flywheel  is  running  steadily  at  the  rate  of  450  revolutions  per 
minute.  What  angle  does  one  of  its  spokes  generate  in  2  sec?  In  1.2 
sec? 


*  Either  of  these  directions  may  of  course  be  chosen  as  the  positive  direc- 
tion of  rotation,  the  other  is  then  the  negative  direction.  The  choice  here 
made  is  the  customary  one  for  angles ;  but  in  many  kinds  of  machinery,  the 
other  sense  of  rotation  is  considered  positive,  as  in  the  case  of  a  clock. 


84  PLANE  TRIGONOMETRY  [IX,  §  59 

3.  Find  the  sum,  or  resultant,  of  two  forces  that  act  in  the  same  line 
whose  intensities  (measured  in  pounds)  are  —  6  and  +  10,  respectively. 
Draw  a  figure  to  represent  the  solution. 

4.  If  three  forces  of  intensities  +7,  —  16,  -f  2  (lb.),  respectively, 
act  on  a  body  in  the  same  line,  find  the  resultant  force.  Draw  a 
figure. 

5.  If  a  man  walks  with  a  speed  of  4  mi.  per  hour  toward  the  rear  of 
a  train  going  35  mi.  per  hour,  find  his  actual  speed.     Draw  a  figui-e. 

6.  A  man's  gains  and  losses  (indicated  by  — )  in  business  in  succes- 
sive months  are  |260,  —  $118,  |35,  |712,  —  |15.  Find  the  total  gain 
and  the  average  gain  per  month.     Draw  a  figure. 

7.  By  means  of  a  ruler  and  a  protractor,  construct  the  following 
angles  and  their  sums  ;  check  by  adding  their  numerical  measures. 

(a)  _  76°  and  125°.         (6)  66""  and  -  30°.        (c)  45°  and  30°,  and  70° 
(d)  -  60°  and  -  36°.      (e)  485°  and  55°.  (/)   -  750°  and  30°. 

8.  With  some  two  of  the  angles  just  given  verify  a  -f  j3  =  ^  +  a. 

9.  (a)  Construct  27°  +  85°  +  (  -  45°)  +  135°. 
(6)  Construct  -  150°  +  96°  +  24°  +  (-  80°). 

10.  If  a  wheel  is  rotating  120°  per  second,  how  many  revolutions  does 
it  make  per  minute  ?  how  many  per  hour  ?  How  many  degrees  does  it 
turn  through  per  minute  ? 

11.  Express  an  angular  speed  of  2.5  revolutions  per  second  in  degrees 
per  second  ;  in  revolutions  per  minute  ;  in  degrees  per  minute. 

12.  A  flywheel  rotates  at  the  rate  of  40  revolutions  per  minute. 
Through  what  angle  does  one  of  its  spokes  turn  in  a  second  ? 

13.  Eeduce  an  angular  speed  of  3.4  revolutions  per  second  to  degrees 
per  second  ;  to  degrees  per  minute  ;  to  revolutions  per  minute. 

14.  Find  the  angular  speed  of  the  rotation  of  the  earth  on  its  axis 
(a)  in  revolutions  per  minute  ;  (6)  in  degrees  per  second. 

15.  Construct  a  right  triangle  whose  sides  are  3  and  4  ;  construct  an 
angle  which  is  3  times  the  smaller  angle  of  this  triangle. 

16.  Construct  the  following  angles  and  place  them  on  the  axes, 
(a)   -  150°  ;    (6)  285°  ;   (c)  480°  ;    (d)  670°  ;    (e)   -  225°  ;  (/)   -  450°. 

17.  In  what  quadrant  is  each  of  the  following  angles  :  459°,  682°,  725°, 
-  100°,  -  1090°,  ±  85°,  ±  95°,  ±  175°,  ±  185°,  di  265°,  ±  276°,  ±  355°  ? 

18.  Taking  a  =  60°,  i3=-300°,  7=-50^  5  =r  310°  draw  a  figure 
showing  that  a  differs  from  /3,  and  also  that  y  differs  from  5  by  360°. 

19.  Find  the  angle  between  0°  and  360°  which  differs  from  each  of  the 
following  angles  by  a  multiple  of  360°  : 

(a)   -42°  13';  (b)  -  842°  j  (c)  364°  23';  (d)  2700°. 


EX,  §621  RADIAN  MEASURE  85 

60.  Measurement  of  Angles.  An  angle  may  be  named  and 
used  belore  it  is  expressed  in  any  system  of  measurement. 
Thus,  we  may  refer  to  an  angle  ^  of  a  right  triangle  whose 
perpendicular  sides  are  16  in.  and  24  in.,  respectively ;  and 
we  can  compute  tan  A  =  24/16  =  1.5,  etc.,  without  measuring 
A  in  terms  of  any  unit  angle.  General  theorems  like  the  law 
of  sines  remain  true  in  any  system  of  measurement. 

The  unit  angle  (see  §  2)  chiefly  used  in  Geometry  and 
Trigonometry  is  the  degree  with  its  subdivisions  minute,  tenth 
of  minute,  second,  with  which  the  student  is  familiar.  It  is 
often  convenient  to  use  another  unit  angle  called  the  radian, 

61.  Radian  Measure  of  Angles.  =^  A  radian  is  a  positive 
angle  such  that  when  its  vertex  is  placed  at  the  center  of  a 
circle,  the  intercepted  arc  is  equal  in  length  to  the  radius. 

This  unit  is  thus  a  little  less  than  one  of 
the  angles  of  an  equilateral  triangle ;  in 
fact  it  follows  from  the  geometry  of  the 
circle,  since  the  length  of  a  semicircum- 
ference  is  rrr,  that 

(1)  TT  rac/zans  =  180°,  where  17  =  3.14159,  F^^i. 
whence   1   radian  =  57°  17'  44".806,   or  57°.3  approximately. 

It  is  easy  to  change  from  degrees  to  radians  and  vice  versa 
by  means  of  relation  (1),  which  should  be  remembered.  Con- 
version tables  for  this  purpose  are  printed  in  Tables,  pp.  91-93. 

62.  Use  of  Radian  Measure.  It  is  shown  in  geometry  that 
two  angles  at  the,  center  of  a  circle  are  to  each  as  their  inter- 
cepted arcs  ;  therefore  if  an  angle  at  the  center  is  measured  in 
radians  and  if  the  radius  and  the  intercepted  arc  are  measured 
in  terms  of  the  same  linear  unit,  their  numerical  measures 
satisfy  the  simple  relation : 

(2)  arc  =  angle  x  radius. 

*  Sometimes  also  called  circular  measure. 


86  PLANE  TRIGONOMETRY  [IX,  §64 

In  other  words,  the  number  of  linear  units  in  the  arc  is 
equal  to  the  product  of  the  number  of  radians  in  the  angle  by 
the  number  of  linear  units  in  the  radius. 

Example  1 .    Find  the  difference  in  latitude  of  two  places  on  the  same 
meridian  200  mi.  apart,  taking  the  radius  of  the  earth  as  4000  mi. 
Angle  =  arc/radius  =  1/20  in  radians  =  2°  51'  63",  approximately. 

63.  Angular  Speed.  In  a  rotating  body  a  point  P,  which  is 
at  a  distance  r  from  the  axis  of  rotation,  moves  through  a  dis- 
tance 2  7rr  during  each  revolution  or  through  a  distance  r  while 
the  body  turns  through  an  angle  of  one  radian.  Therefore  if  v 
is  the  linear  (actual)  speed  of  P  (in  linear  units  per  time  unit, 
e,g,  feet  per  second),  and  if  w  is  the  angular  speed  of  the  ro- 
tating body  (in  radians  per  time  unit,  e,g,  radians  per  second), 
then  their  numerical  measures  satisfy  the  relation 

(3)  i;  =  r  .  CD ; 

hence  the  angular  speed  of  a  rotating  body  is  numerically  equal 
to  the  actual  speed  of  a  point  one  unit  from  the  axis  of  rotation. 
Engineers  usually  express  the  angular  speed  of  the  rotating 
parts  of  machinery  in  revolutions  per  minute  (R.  P.  M.)  or 
revolutions  per  second  (R.  P.  S.).  These  are  easily  reduced  to 
radians  per  minute  (or  per  second)  by  remembering  that  one 
revolution  equals  2  tt  radians. 

Example  1.  A  flywheel  of  radius  2  ft.  rotates  at  an  angular  speed  of 
2.5  R.  P.  S.     Find  the  hnear  speed  of  a  point  on  tlie  rim. 

In  radians  per  second,  w  =  2.6x27r  =  5  7r,  and  for  a  point  2  ft.  from 
the  axis  of  rotation  v  ~2  x  ^ir  =  31.416  ft.  per  second. 

Example  2.  Find  the  angular  speed  of  a  34-inch  wheel  on  an  auto- 
mobile going  20  mi.  per  hour. 

Every  time  the  wheel  turns  through  a  radian  the  car  goes  forward  17 
in.  (the  length  of  the  radius),  and  20  mi.  per  hour  =  362  in.  per  second  ; 
therefore  the  wheel  turns  through  352/17  =  20.7,  radians  per  second. 

64.  Notation.  In  measuring  angles  in  radian  measure  we 
shall  adopt  the  practice  universal  in  advanced  work  and  write 
only  the  numerical  measure  of  the  angle  in  terms  of  the  unit 


IX,  §  64]  RADIAN  MEASURE  87 

one  radian.  Thus  in  the  expression  tan  x,  the  letter  x  will  de- 
note a  number  (the  numerical  measure  of  an  angle)  rather  than 
the  angle  itself.     See  §  2. 

When  necessary,  to  call  attention  to  the  fact  that  radian 
measure  is  intended,  the  symbol  (^''^)  is  appended  to  the  nu- 
merical measure,  thus  : 

1(^>  =  1  radian  =  57°  17'  44".8, 
2<'">  =  2  radians  =  114°  35'  29".6, 
TT^'^^  =  TT  radians  =  180°  =  2  rt.  A, 
(7r/2)('>  =  7r/2  radians  =  90°  =  1  rt.  Z, 

and  so  forth. 

As  it  happens  that  the  acute  angles  whose  trigonometric  functions  are 
most  easily  recalled  without  consulting  tables  are  simple  fractional  parts 
of  180°,  the  number  tt  often  appears  as  a  factor  of  the  numerical  measure 
of  angles.  In  this  system,  for  example,  sin  (7r/2)  =  1,  cos  (tt/S)  =  1/2, 
tan  (7r/4)  =  1,  etc. 

The  use  of  pure  numbers,  such  as  2  or  tt  in  place  of  an  angle  is  pre- 
cisely similar  to  the  use  of  10  for  10  feet  or  10  inches  in  expressing  lengths. 
The  student  should  supply  the  unit  of  measurement  (radians  or  feet  or 
inches),  and  should  not  confuse  the  number  tt  (=  3.14159  •••)  with  the 
angle  whose  measure  is  tt  radians^  as  he  should  not  confuse  the  number 
10  with  the  distance  10  feet. 

EXERCISES  XXIII.  — ANGULAR  SPEED  — RADIAN  MEASURE 

1.  Express  the  following  angles  in  degrees,  minutes,  and  seconds : 

(a)  7r(^)/4;         (6)  7rW/6  ;         (c)  2  7r(»-)/3;         (d)  3K 

2.  Express  the  following  angles  in  radians  : 

(a)  25°;         (&)  30°  ;         (c)  35°  ;         ((2)  28°  39';         (e)  114°  36'. 

3.  How  far  short  of  one  revolution  is  6(»")  ? 

4.  To  gain  ability  to  judge  the  size  of  angles  in  circular  measure, 
express  approximately  (to  within  1°)  angles  whose  sizes  are  l^*"),  4('"),  5(»">, 
S^*").  Draw  an  angle  which  is  about  your  impression  of  an  angle  of  2(*">, 
and  measure  it  with  a  protractor.     Do  not  revise  your  figures. 

5.  If  a  vehicle  moves  at  the  rate  of  15  ft.  per  second,  through  what 
angle  does  one  of  its  wheels,  3  ft.  in  diameter,  revolve  in  1  sec.  ? 

Ans.   IOC'). 


88  PLANE  TRIGONOMETRY  [IX,  §  64 

6.  If  the  linear  speed  of  a  vehicle  is  30  mi.  per  hour,  what  is  the 
angular  speed  of  one  of  its  wheels  which  is  4  ft.  in  diameter  ? 

Ans.   22  radians  per  second. 

7.  A  wheel  6  ft.  in  diameter  is  connected  by  a  belt  40  ft.  in  length 
with  a  wheel  4  ft.  in  diameter.  If  the  large  wheel  makes  30  revolutions 
per  minute,  how  often  does  the  seam  of  the  belt  pass  this  wheel  ?  What 
is  the  angular  speed  of  the  smaller  wheel? 

Ans.   6y^  sec,  3}|  radians  per  second. 

8.  Find  the  angular  distance  on  the  earth  between  two  points  whose 
distance  from  each  other,  on  the  arc  of  a  great  circle,  is  800  miles. 
[Take  the  radius  of  the  earth  to  be  4000  miles.  ]  Ans.    11°  21'  33". 

9.  Find  the  distance  in  miles  between  two  points  on  the  earth's  sur- 
face whose  angular  distance  is  1° ;  between  two  points  whose  angular 
distance  is  0.25  radians.  Ans.   69.81,1000. 

10.  Find  the  length  of  the  subtended  arc  of  an  angle  of  3.46  radians 
at  the  center  of  a  circle  of  radius  5.  Ans.    17.3 

11.  Find  the  length  of  the  subtended  arc  of  an  angle  of  55°  at  the 
center  of  a  circle  of  radius  3.  Ans.   2.8798 

12.  Find  the  angle  at  the  center  which  subtends  an  arc  of  3  ft.  on  a 
circle  of  radius  4  ft.  Express  the  angle  in  radians  and  in  degrees,  and 
compare  the  work  done  in  the  two  cases.  Ans.   |  radian  =  42^.97+ 

13.  Reduce  to  radian  measure  by  means  of  Tables  IV,  p.  91  : 
(a)  23°  40'  ;     (6)  68°  45'  20"  ;     (c)  138°  35'  15". 

Ans.  0.4130612,     1.2000109,     2.4188082 

14.  Reduce  to  degree  measure  by  means  of  the  Tables  pp.  92-93  : 
(a)  3.46W  ;         (b)  .256(»-) ;         (c)   .0127(^)  ;         (d)  8.240-). 

Ans.    198°  14'  36".2,  14°  40'  3".8,  43'  39".5,  472°  7'  2" 

15.  Reduce  the  following  angular  speeds  to  degrees  per  second  ;  to 
revolutions  per  second  ;  to  revolutions  per  minute  : 

(a)  4.5(*">  per  sec.  ;        (6)  2.48^^)  per  sec.  ;        (c)  10.54(*">  per  sec. 
Ans.    (a)  257.83,   0.7162,   42.972  ;  (6)   142.09,   0.3947,   23.682; 

(c)  603.90,   1.6775,   100.66 


CHAPTER   X 


FUNCTIONS   OF  ANY  ANGLE 

65.  Resolution  of  Forces.    Projections.    In  §  26,  p.  34, 

we  saw  how  to  find  the  components  of  a  force,  or  a  velocity, 
on  any  line,  as  the  projection  of  the  force  on  that  line ;  and  we 
saw  that  the  components  of 

y 


f:(x.y) 


a  force  F  on  each  of  two  per- 
pendicular axes,  even  when 
the  angle  a  is  obtuse,  are 

(1)  F^  =  'PY0J^F=FG08a, 
Fy  =  Pro j  ^  F  =  i^  sin  a.  Fig.  82. 

If  several  forces  occur  in  the  same  problem,  some  of  them 
may  make  an  angle  a  greater  than  180°  with  the  positive 
direction  OX.  It  is  convenient  to  define  cos  a  and  sin  a  for 
angles  greater  than  180°  so  that  the  equations  (1)  remain  true. 
If  we  do  so,  the  projection  on  the  two  axes  of  any  directed 
segment  of  length  r  joining  the  origin  0  to  a  point  P  are 

(2)  X  =  Proj^  r  =  r  cos  a,        y  =  Proj^  r  =  r  sin  a, 

where  a  is  the  angle  between  the  positive  direction  OX  and  the 
positive  direction  OP,  and  may  be  an  angle  of  any  size,  posi- 
tive or  negative.    Hence  the  desired  definitions  are : 

(3)  cosa  =  -,        sma  =  -- 

These  definitions  are  consistent  with  those  already  given, 
§§  11,  35,  for  the  sine  and  the  cosine ;  i.e.  in  case  0°^  a  ^  180°, 
they  determine  the  same  values  as  the  earlier  definitions. 

66.  General  Definitions.  Trigonometric  Functions  of  Any 
Angle.     The  definitions  of  sin  a  and  cos  a  given  in  §  65  have, 


90 


PLANE  TRIGONOMETRY 


P^,§66 


of  course,  no  necessary  dependence  upon  forces.  Each  is  a 
number  which  depends  only  on  the  magnitude  and  sign  of  the 
angle.  A  purely  geometric  definition  of  these  and  of  the 
other  trigonometric  functions  of  any  angle  a,  consistent  with 

the  definitions  of  §§  11,  35,  and  with 
the  fundamental  relations  between 
them,  such  as  tan  a  =  sin  a/cos  a, 
sin^  a  -\-  cos2  oc  =  1,  the  reciprocal  re- 
lations, etc.,  may  be  made  as  follows : 
Place  the  given  angle  on  a  pair 
of  rectangular  axes,  and  select  any 
^^'     '  point  P  whose  coordinates  are  {x,  y) 

terminal  side  at  a  distance   r  >  0  from   the  origin. 


P  (x.y) 


P  (x.y) 


on   the 
Then 

(4) 
(5) 
(6) 
(J) 
(8) 
(9) 


sina  = 


cos  a  =  -  = 
r 


y  __  ordinate 
r  ""  radius  ' 
abscissa 


radius  ' 

y      ordinate 

tan  a  =  -  =  -^ — ; — 

X     abscissa 

X     abscissa 
ctn  a  =  -  =  — -T-. — -r~  > 
y      ordinate 

r       radius 
sec  a  =  -  =  -^j — -. — , 
X     abscissa 


CSC  a  =  -  = 


radius 


y     ordinate ' 


provided  x4^()\* 
provided  ?/  =5^  0  ; 
provided  a;  ^^fc  0 ; 
provided  y  ^0. 


Three  additional  functions  sometimes  used  are : 
(10)  The  versed  sine  of  a :  vers  oc  =  1 

(11) 

(12) 

and  also  the  coversed  sine  of  a  =  1  —  sin  a. 


cos  a. 

The  haver  sine  of  a :  hav  a  =  ^(l  —  cos  a). 
The  external  secant  of  a  :  exsec  a  =  sec  a  - 


*  The  exceptions  noted  are  based  on  the  general  principle  that  a  fractional 
expression  does  not  represent  a  number  if  its  denominator  is  zero. 


X,  §  67]  FUNCTIONS  OF  ANY  ANGLE  91 

By  these  definitions  every  angle  has  a  sine  and  a  cosine,  be- 
cause in  the  ratios  yjr  and  xjr  the  denominator  r  is  never  zero. 
There  is  no  secant  or  tangent  ^  for  90°,  or  for  270°,  or  for  any 
angle  whose  terminal  side  coincides  with  either  the  positive  or 
negative  end  of  the  ^/-axis,  because  the  denominator  x  in  the 
ratios  r/o;,  ylx^  is  zero.  Similarly,  there  is  no  cosecant  or  co- 
tangent for  0°  or  for  180°,  or  for  any  angle  whose  terminal 
side  coincides  with  the  positive  or  negative  end  of  the  ic-axis. 
There  exists  a  tangent,  cotangent,  secant,  and  cosecant  for 
every  angle  except  those  just  mentioned. 

If  two  angles  differ  by  any  multiple  of  360°  it  is  evident 
that  any  one  of  the  trigonometric  functions  will  have  the  same 
value  for  both  of  them  because  the  initial  sides  of  the  two  angles 
(when  placed  on  the  axes)  will  coincide,  and  also  their  terminal 
sides.  It  follows  that  for  a  point  P  on  the  common  terminal 
side  the  values  of  a?,  ?/,  and  r  are  the  same  for  both  angles ; 
hence  the  ratio  which  defines  any  given  function  will  be  the 
same  for  both  angles. 

For  example  :  sin  (-  295°)  =  sin  ^h"",  cos  (-  315°)  =  cos  45°, 
tan  1476°=  tan  36°,  sin  ((9  -  180°)  =  sin  (180°+  ^),  cos  (x  -  90°) 
=  cos  (270°  +  x),  tan  (360°  -  y)=  tan  (-  y). 

67.  Algebraic  Signs  of  Trigonometric  Functions.  The  sine 
of  any  angle  in  the  first  or  second  quadrant  is  positive,  because 
the  ordinate  of  any  point  above  the  aj-axis  is  positive ;  the  sine 
of  any  angle  in  the  third  or  fourth  quadrant  is  negative,  be- 
cause the  ordinate  of  any  point  below  the  a;-axis  is  negative. 

The  cosine  of  any  angle  in  the  first  or  fourth  quadrant  is 
positive,  because  the  abscissa  of  any  point  to  the  right  of  the 


*  To  say  that  90°  has  no  tangent  does  not  mean  that  the  tangent  of  90°  is 
zero.  When  we  say  that  an  article  has  no  value  we  mean  that  it  has  a  value 
and  that  value  is  zero.  Not  so  here.  Since  the  general  definition  of  tangent 
4oes  not  apply  to  90°,  we  could,  if  we  found  it  convenient,  define  tan  90°,  but 
we  do  not ;  we  leave  it  undefined.  Often  it  is  said  tan  90°  =  oo  ,  but  this  does 
not  mean  that  90°  has  a  tangent ;  it  means  that  as  an  angle  a  increases  from 
0°  to  90°,  tan  a  increases  without  limit,  and  that  before  a  reaches  9(P. 


92 


PLANE  TRIGONOMETRY 


[X,§67 


2/-axis  is  positive ;  similarly,  the  cosine  of  any  angle  in  the 
second  or  third  quadrant  is  negative. 

Similarly,  the  signs  of  tan  a,  ctn  a,  sec  a,  esc  a,  etc.,  may  be 
determined  directly  from  a  figure;  they  are  as  follows  : 


Quadrant 

sin  a 

cos  a 

tana 

etna      I 

sec  a 

CSC  a 

1st 

+ 

+ 

+ 

+ 

+ 

+ 

2d 

+ 

- 

- 

- 

- 

+ 

3(1 

- 

- 

+ 

+ 

- 

- 

4th 

- 

+ 

- 

- 

+ 

- 

Note.     (1)  tana  is  positive  (negative)  v^hen  sin  a  and  cos  a  have  hke 
(unhke)  signs  ;  (2)  reciprocals  have  the  same  sign. 


EXERCISES  XXIV.  — FUNCTIONS  OF  THE  GENERAL  ANGLE 

1.  By  placing  the  angles  on  the  axes,  show  from  the  definitions  that 
(  a)  sin  225°  =  -  V2/2,  cos  225°  =  -  \/2/2. 

(  6  )  sin  150°  =  1/2,  cos  150°  =  -  \/3/2. 
(c  )  sin  330°  =  -  l/2,_cos  330°  =  V3/2. 
(d)  sin  (-  315°)  =  V2/2,  cos  (-  315°)  =  V2/2. 
(  e)  sin  (-  1020°)  =  V3/2,  cos  (-  1020°)  =  1/2. 
(/)  sin  180°  =  0,  sin  (n  •  180°)  =  0  ;  for  n  =  ±  1,  ±  2,  ±  3,  .... 
.      {g)  cos 90°  =  0,  cos[(2n-  1)90°]=  0;  forn=±  1,  ±2,  ±3,  .... 

2.  Which  of  the  following  are  positive  and  which  negative  ?  sin  72°, 
sin  352°,  sin  850°,  tan  128°,  sec  260°,  sin  (-20°),  cos  (-380°),  sin  (-260°),. 
cos  160°,  ctn 280°,  cos 33°,  csc91°,  cos  (-  40°),  tan  (-  140°),  cos(-400°). 

3.  Prove  for  any  angle  a  that  sin2  a  -f  cos2  a  =  1.    [Use  x"^  +  y^  =  r2.] 
Prove  each  of  the  other  P3rthagorean  relations  for  any  angle  a  : 

1  +  tan2  a  =  sec2  a,  if  cos  cc  :^  0  ;  1  -|-  ctn2  a  =  csc2  a,  if  sin  a  =^  0. 

4.  Prove  that  ctn  a,  sec  a,  esc  a  are  the  reciprocals  of  tan  a,  cos  a, 
sin  a,  respectively,  for  all  values  of  a  for  which  both  are  defined. 

6.  (a)  Prove  that  the  sine  of  any  angle  in  the  first  or  second  quad- 
.  rant  is  between  0  and  1.  (6)  Prove  that  the  cosine  of  any  angle  in  the 
1st  or  4th  quadrant  is  between  0  and  1. 

6.  Prove  that  if  an  angle  is  not  an  odd  multiple  of  a  right  angle  its 
sine  is  between  —  1  and  +  1 ;  and  conversely.  For  what  angles  is  sin  a 
=  +  1 ;  sin  a  =  —  1 ;  cos  a  =  +  1  ? 


X,  §68] 


FUNCTIONS  OF  ANY  ANGLE 


93 


7.  Show  that  tan  a  =  sin  cc/cos  a  for  all  values  of  a,  if  cos  a  ^  0. 

8.  Show  that  tan  a  and  ctn  a  may  have  any  values  whatever, 

9.  Show  that  vers  a  and  hav  a  are  always  positive  or  zero, 

10.    If  an  angle  a  starts  at  0^  and  gradually  increases  to  360°,  show 
that  the  behavior  of  sin  a  and  cos  a  will  be  as  indicated  in  this  table  : 


a 

0° 

0°<-<90° 

90° 

90°<a<180° 

180° 

180°<a<270° 

270° 

270°<a<360° 

360'' 

sin  a 

0 

increases  to 

1 

decreases  to 

0 

decreases  to 

-1 

increases  to 

0 

cos  a 

1 

decreases  to 

0 

decreases  to 

—  1 

increases  to 

0 

increases  to 

1 

11.    By  placing  the  angles  indicated  on  rectangular  axes  determine  the 
numbers  to  fill  the  blanks  in  the  following  table  : 


a 

30° 

45° 

60° 

120° 

135° 

160° 

210° 

225° 

240° 

300° 

315° 

330° 

since 

cos  a 

12.  Assuming  that  the  sun  passes  directly  overhead,  trace  the  change 
in  the  length  of  the  shadow  of  an  object  from  dawn  to  sunset.  Which 
trigonometric  function  do  you  think  of  in  this  problem  ? 

13.  Assuming  the  results  of  Exs.  10  and  11,  derive  from  them  the 
variation  of  the  tangent  from  0^  to  360°  and  its  values  at  each  of  the 
angles  mentioned  in  Ex.  11.     Do  the  same  for  ctn  a,  sec  a,  esc  a. 

68.   Reading  of  Tables.     Sine  and  Cosine  of  —9  and  90° +  9. 

In  order  to  find  the  value  of  any  one  of  the  trigonometric  func- 
tions of  a  given  angle  we  consult  the  tables.  In  the  tables  the 
values  of  the  different  functions  are  printed  only  up  to  45°. 
To  find  the  sine  of  an  acute  angle  greater  than  45°  we  make 
use  of  the  relation  sin  a  =  cos  (90°  —  a).  The  tables  are  ar- 
ranged to  facilitate  this  by  having  the  angles  above  45°  printed 
at  the  bottom  of  the  page,  and  the  column  headings  changed 
from  sine  to  cosine,  etc.     (See  Tables,  p.  22.) 

If  we  wish  to  find  the  sine  of  an  angle  greater  than  90°,  we 
must  find  a  way  to  express  the  sine  in  terms  of  some  function  of 


94 


PLANE  TRIGONOMETRY 


[X,§68 


K(-b.a}_ ,H  (b,a) 


an  acute  angle.  We  proceed  to  find  expressions  for  the  values 
of  the  sine  and  the  cosine  of  the  angles  90°  ±  0,  180°  ±  0, 
270°  ±  e,  and  360^  -  0. 

To  construct  these  angles  we  draw  a  circle  of  radius  r  with 
its  center  at  the  origin  and  draw  the  diameters  HN,  KS  mak- 
ing the  angle  0  with  the 
2/-axis  to  the  right  and  left, 
and  also  the  diameters  PM, 
TL  making  the  angle  6  with 
the  a>-axis  above  and  below. 
The  angles  XOH,  XOK; 
XOL,  XOM)  XON.XOS', 
and  XOT,  are  the  angles 
mentioned  above,  and  XOP 
is  the  angle  0.  Denote  the 
coordinates  of  the  point  P 
by  (a,  h)  ;  then  because  the 
triangle  OAH  is  congruent 
to  the  triangle  OOP  the  coordinates  of  the  point  H  are  (6,  a), 
and  in  the  same  way  the  coordinates  of  the  points  K,  i,  M, 
N,  S,  T  are  easily  seen  to  be  as  indicated  in  the  figure.  We 
are  now  able  to  read  off  the  values  of  the  trigonometric  func- 
tions of  the  various  angles  from  the  figure,  in  terms  of  a,  6, 
and  r ; 
thus  sin  0  =  6/r, 

cos  (90°  -0)==  h/r,  cos  (90°  +  6>)  =  -  h/r, 

sin  (180°  -0)= h/r,  sin  (180°  +  (9)  =  -  h/r, 

cos  (270°  -  ^)  =  -  6/r,  cos  (270°  +  (9)  =  6/r, 

sin  (360°  -  ^)=  sin  (-  6>)=  -  h/r. 
Hence  we  have 

cos  (90°  -  ^)  =  sin  0  cos  (90°  +  ^)  =  -  sin  ^ 

sin  (180°-  0)  =  sin  6  sin  (180°  +  ^)  =  -  sin  ^ 

cos  (270°  -f  ^)  =  sin  0  cos  (270°  -  /)  =  -  sin  0 

sin  (360°  -  ^)  =  sin (-  ^)  =  -  sin  6 


X,  §  69]  FUNCTIONS  OF  ANY  ANGLE  95 

Similarly  we  obtain  from  the  figure  cos  0  =  a/r, 

sin  (90°  -6)=  a/r,  sin  (90° + ^)  =  a/r, 

cos  (180°-  6>)  =  -  a/r,  cos  (180°  +  ^)  =  -  a/r, 

sin  (270°  -6)  =  -  a/r,  sin  (270°  +  (9)  =  -  a/r, 

cos  (360°  -  ^)=  cos  (-  e)=a/r. 
Whence, 

sin  (90°  -0)=  cos  (9  cos  (180°  -  ^)  =  -  cos  ^ 

sin  (90°  +  6)=  cos  6  cos  (180°  +  ^)  =  -  cos  ^ 

cos  (860°  -  ^)  =  cos  (  -  6)  sin  (270°  _  ^)  =  _  cos  d 

=  cos  ^  sin  (270°  +  (9)  =  -  cos  ^ 

These  formulas  together  with  the  fact  mentioned  in  §  66, 
that  a  function  of  an  angle  a  has  the  same  value  as  the  same 
function  of  any  angle  that  differs  from  cc  by  a  multiple  of 
360°,  are  sufficient  to  enable  one  to  find  the  value  of  any  one 
of  the  functions  of  any  angle  from  the  tables.* 

Example  1.     Find  the  sine  of  793°  22'. 

The'  angle  73°  22'  differs  from  the  given  angle  by  720°,  which  is  a 
multiple  of  360° ;  hence  the  required  value  is  the  same  as  sin  73°  22'. 
From  the  tables  this  value  is  found  to  be  .95816 

69.  Solution  of  Trigonometric  Equations.  We  are  now  able 
to  give  the  general  solutions  of  the  equations  sin  0  =  c  and 
cos  0  =  G  where  c  is  any  number  lying  between  +  1  and  —  1. 
In  the  first  place,  it  is  clear  that  there  are  two  and  only  two 
angles  between  0°  and  360°  which  will  satisfy  either  of  these 
equations.  For  in  the  above  figure  there  are  only  two  points 
of  the  circle  for  which  x  has  a  given  value  between  +  r  and 
—  r,  and  likewise,  only  two  points  for  which  y  has  a  given 
value  between  +  r  and  —  r ;  a  radius  drawn  to  either  of  these 
two  points  will  be  the  terminal  side  of  an  angle  between  0°  and 
360°,  satisfying  the  first  equation  if  y  is  chosen  so  that  y/r  =  c 
and  satisfying  the  second  if  x  is  chosen  so  that  x/r  =  c.  To 
obtain   the   general   solution   we  add   or  subtract  any  whole 

*  The  proofs  given  above  are  for  the  case  in  which  0  is  an  acute  positive 
angle.    The  formulas,  however,  are  true  for  any  value  of  0  whatever. 


96  PLANE  TRIGONOMETRY  [X,  §  70 

multiple  of  360°  to  either  of  the  solutions  just  found.  The 
solutions  which  lie  between  0°  and  360°  can  be  found  from  the 
tables  by  means  of  the  formulas  given  above. 

70.  Illustrative  Examples  on  Composition  and  Resolution 
of  Forces. 

Example  1.  Find  the  components,  Rj.t  -Ry »  of  the  resultant  of  two 
forces,  the  first  of  12  lb.  acting  at  an  angle  of  30°  with  the  horizontal,  the 
second  of  20  lb.  acting  at  an  angle  60°  with  the  horizontal. 

Solution.  To  solve  this  we  make  use  of  the  principle  that  the  projec- 
tion on  any  line  of  the  resultant  of  any  number  of  forces  is  the  algebraic 
sum  of  the  projections  of  the  component  forces. 

By  equations  (1),  §  65,  the  horizontal  component  of  the  first  is 
12  cos  30°,  and  of  the  second,  20  cos  60°  :  hence 

B^  =  12  cos  30°  4-  20  cos  60°  =  10.392  +  10.000  =  20.392 
In  a  similar  manner  we  find 

By  =  12  sin  30°  +  20  sin  60°  =  6.000  +  17.320  =  23.320 

We  can  easily  find  the  magnitude  of  the  resultant  from  the  equation 

B2  =  E2^  +  B\  =  (20.392)2  +  (23.320)2  =  959.665 

Hence  

12  =\/(959.665)=  30.979 

The  direction  of  the  resultant  is  given  by  the  equation 

tan^  =  By-^  B^  =  23.320  --  20.392  =  1.1436 
Hence 

^  =  48°50^ 

Example  2.  Find  the  magnitude  and  the  direction  of  the  resultant  of 
the  two  forces  F  =  (17,  128°),  G  =(24,  213°). 

[Note.  The  notation  (24,  213°)  means  a  force  of  magnitude  24  acting 
at  an  angle  of  213°  with  the  positive  ic-axis.] 

The  method  of  solution  is  the  same  as  in  Example  1  ;  we  find 

F^  =  17  cos  128°  =  -  17  sin  38°  (by  §  68). 

G^  =  24  cos  213°  =  -  24  cos  33°  (by  §  68) . 

TTpticp 

i?,  =  -  17  sin  38°  -  24  cos  33°  =  -  10.466  -  20. 128  =  -  30.594 

Similarly  we  obtain 

By  =  17  cos  38° -24  sin  33°  =  13.396  -  13.071  =  .325 
B  =  V(B\  +  B\)=S0.6n 

0  =  arctan  (.325/-  30.594)  =  arctan  (-  .01062)  =  180°-  36^4  =  179°  23'.6 


X,  §  70]  FUNCTIONS  OF  ANY  ANGLE  97 

EXERCISES  XXV.  — READING  OF  TABLES  —  REDUCTION  TO 
FUNCTIONS  OF  ACUTE  ANGLES 

1.  Express  the  following  as  functions  of  acute  angles  not  greater  than 
45°.     Make  use  of  congruent  angles  whenever  advantageous  : 

(a)  sin  150°  21'.  (6)  cos  125°  15'.  (  c  )  tan  283°  45'. 

(d)  ctn(--36°16').        (e)  sec460^  (/)  esc  (- 210° 20'). 

(g)  sin(-943''24').       (/t)  cos55P23'.  (i)  tan  (- 546° 28'). 

2.  From  the  tables  find  the  values  of  the  following  logarithms-, 
(a)  log  (-  cos  161°  11').  (6)  log  sin  161°  11'. 

(c)  log  (-sin  217°  17').  (d)  log  (- cos  252°  480- 

[Note  that  the  numbers  in  parentheses  in  (a),  (c),  and  (d)  are  posi- 
tive ;  if  the  minus  sign  were  absent,  each  of  them  would  be  negative. 
Negative  numbers  have  no  real  logarithms.  ] 

3.  Compute  the  values  of  the  following  expressions  by  logarithms  : 
(a)  2.35  sin  148°  23'.         (6)  24.8  cos  160°  40'.        (c)  16.2  cos  320°  45'. 

4.  Solve  the  following  trigonometric  equations  : 

(a)  cos2 1  —  sin2  t  =  smt. 

Solution.  In  this  equation  cos2  t  may  be  replaced  by  its  equal  1  — 
sin2  i ;  the  equation  then  becomes  a  quadratic  in  sini,  viz.:  2sm'^t  + 
sint—  1  =  0.  This  equation  is  equivalent  to  the  given  one;  i.e.  every 
solution  of  either  is  a  solution  of  the  other.  The  solutions  may  now  be 
found  by  factoring : 

(2sini-  l)(sini4-  1)  =  0. 

Hence  we  have  either  sin  i  +  1  =  0,  whence  sin  t  =  —  1,  and  t  =  270°  or 
t  =  270°  +  k  360°;  or  else  2  sin  i  =  1,  whence  sin  i  =1/2  and  f  =  30°  + 
k  360°  or  t  =  150°  +  k  360°.     There  are  no  other  solutions. 

(6)2  sin2  X  —  cos  X  =  1.  (g)  sec2  x  +  tan  x  =  S. 

(c)  cos2x=sin2x.  (^)  4  sec2  x  +  tan  x  =  7. 

(d)  cos2x  +  5sinx=  3.  (i)  tan x -f- ctn x  =  2. 

(e)  cos 2 X  — sin X  =  1/2.  (j)  sinx  +  3  =  cscx. 
(/)  5  sin X  4-  2  cos^x  =  5.  (k)  sin2 x  cos  x  =  sinx. 

5.  Find  the  resultant  (i?,  6)  of  three  forces  (100,  350°),  (150,  490°), 
(200,  720°),  where  (F,  a)  indicates  a  force  of  magnitude  F  and  direc- 
tion a. 

6.  Find  the  components  on  the  axes  of  a  force  of  magnitude  5.74  lb. 
which  makes  an  angle  of  215°  20'  with  the  positive  end  of  the  x-axis. 

7.  Find  the  magnitude  and  the  direction  of  a  force  whose  components 
on  two  perpendicular  axes  are  F^  =  25.46,  Fy  =  38.72 


CHAPTER   XI 


THE   ADDITION   FORMULAS 


71.  The  Addition  Formtilas.  In  the  reduction  of  certair 
trigonometric  expressions  to  simpler  or  more  convenient  forms 
it  is  sometimes  desirable  to  express  a  trigonometric  functioi 
of  the  sum  or  difference  of  two  angles  in  terms  of  functions  o: 
the  separate  angles  forming  the  sum  or  difference.  Withou 
reflection  the  student  might  think  that  sin  (a  -f  p)  would  b( 
equal  to  sin  a  +  sin  )8  by  analogy  with  the  formula  |^(a  +  6 
=  i  a  + 1-  6,  but  a  trial  of  one  or  two  special  cases  will  sho^ 
this  is  not  always  true;  thus,  sin  (60°  +  30°)  is  equal  to  one 
but  sin  60°  +  sin  30°  is  equal  to  ^V3  -f-  \^  which  is  greater  thai 
one.  In  order  to  find  the  correct  formulas  for  sin  (a  -f  y8)  anc 
cos  {a  +  p)  we  make  use  of  the  theory  of  directed  quantities  as 
explained  in  §§  26,   65,  57,  58,  and  Qf5, 

Suppose  a  force  of  magnitude  A  makes  an  angle  a  with  th( 
positive  a>axis,  while  another  force  of  magnitude  B  makes  ai 
angle  a  +  90°  with  this  axis ;  then  the  resultant  R  oi  A  anc 
B  is  represented  by  the  diagonal  OP  of  the  rectangle  of  whicl 
A  and  B  are  two  sides.  The  ^/-component,  R^  of  this  re 
sultant  is 

(1)  ^^  =  ^sina  +  5sin(a  +  90°) 

=  ^  sin  a  4-  J5  cos  a. 
Similarly,  the  a>component  of  R  is 

(2)  jR,  =  ^  cos  a  -f-  -S  cos  (a  -f  90°) 

=  A  cos  a  —  B  sin  a. 
Now  by  §  65,  Fig.  85. 

(3)  i?,  =  i?cos(a4-iS),         R^  =  Rsm(a-\-p), 
where  /3  is  the  angle  between  A  and  the  resultant  R. 


XI,  §  73]  ADDITION  FORMULAS  99 

Inserting  these  values  in  formulas  (1)  and  (2)  we  find 

(4)  R  sin  (a  +  yS)  =  A  sin  a  -h  J5  cos  a. 

(5)  R  cos  {a  -\-  P)=  A  ao^  a  —  B  sin  a. 
Moreover,  from  the  hgure,  -4  =  jK  cos  ^,  B  =  R  sin  fi. 

Substituting  these  values  in  (4)  and  (5)  and  dividing  through 
by  R  we  finally  obtain  the  formulas 

(6)  sin(a  +  P)=  sin  a  cos  p  +  cos  a  sin  p. 

(7)  cos  (a  +  P)=  cos  a  cos  p  —  sin  a  sin  p. 

It  should  be  carefully  noticed  that,  although  in  the  figure 
the  angles  a  and  fi  are  acute  angles,  the  proof  does  not  at  all 
depend  on  this  fact.  Formulas  (6)  and  (7)  are  therefore  true 
for  all  values  of  the  angles  a  and  p. 

72.  The  Subtraction  Formulas.  It  can  be  shown  in  a  man- 
ner exactly  similar  to  the  preceding  that  we  have  also 

(8)  sin  (a  —  p)=  sin  a  cos  p  —  cqs  a  sin  p. 

(9)  cos  (a  —  p)=  cos  a  cos  p  +  sin  a  sin  p. 

It  is  easy  to  derive  (8)  and  (9)  directly  from  (6)  and  (7),  how- 
ever.    Thus,  if,  in  (6),  we  replace  /S  by  —  ^  we  obtain 

sin  (a—  P)  =  sin  a  cos  (—  /8)-f  cos  a  sin (—  )8), 
or,  since  by  §  68,  cos  (—  /3)=cos  /3  and  sin  ( —  )8)  =  —  sin  )3, 
sin  («  —  /?)=  sin  a  cos  p  —  cos  a  sin  )3, 

which  is  (8).     We  prove  (9)  in  a  similar  manner  from  (7). 

These  formulas  are  also  true  for  all  values  of  the  angles  a 
and  /?.  They  are  examples  of  trigonometric  identities  involv- 
ing two  angles. 

73.  Reduction  of  A  cos  o,±B  sin  a.  Such  expressions  as 
A  cos  a  ±  ^  sin  a  which  appeared  in  formulas  (1)  and  (2)  of 
the  previous  article  arise  in  various  connections  ;  for  example, 
a  combination  of  two  vibrations  gives  rise  to  such  a  form. 

It  is  possible,  and  often  convenient,  to  reduce  such  expres- 
sions to  the  product  of  a  single  numl>er,  and  the  sine  (or  the 


100  PLANE  TRIGONOMETRY  \X1,  §  71 

cosine)  of  the  sum  of  two  angles.  The  method  depends  oi 
formulas  (6)  and  (7)  and  upon  the  fact  that  any  two  number! 
are  proportional  to  the  sine  and  the  cosine  of  some  angle. 

Example  1.     Express  3  cos  a  +  4  sin  a  in  the  form  k  sin  (a  +  j3). 
To  solve  this  we  first  find  an  angle  whose  sine  and  cosine  are  propor 
tional  to  3  and  4.     We  may  clearly  choose  an  angle  /3  so  that  sin  )3  =  | 
and  cos  jS  =  |;  hence  we  may  write 

3  cos  Of  +  4  sin  a  =  5(f  cos  a  +  |  sin  a) 

=  5(sin  j8  cos  a  +  cos  /3  sin  a). 
Hence  by  formula  (6)  we  have 

3  cos  a  +  4  sin  a  =  5  sin  (/3  4-  a). 
From  the  tables  ^  =  36°  62'. 

EXERCISES   XXVI.  — ADDITION   FORMULAS 

1.  Given  sin  a  =  3/5,  sin  ^  =  5/13  ;  find  sin  (a  -^  ^). 

(a)  When  a  and  /3  are  both  acute  ;  (6)  when  a  and  jS  are  both  obtuse 

2.  Find  sin  (45°  +  x),  cos  (45°  +  x),  sin  (30°  +  x),  cos  (30°  +  x)   ii 
terms  of  sin  x  and  cos  x. 

3.  Given  that  x  and  y  are  both  obtuse  angles  and  that  sin  x  =  1/2 
sin  2/  =  1/  3  ;  find  sin  (x  +  y)  and  cos  (x  -\-y). 

4.  Use  the  addition  formulas  to  express  sin  (90°+  a)  and  cos  (90°+ a 
in  terms  of  sin  a  and  cos  a. 

5.  Prove  that  sin  (60°  +  x)  —  cos  (30°  +  x)  =  sin  x. 

6.  Express  sin  (ot  +  /3  +  ^)  in  terms  of  sines  and   cosines  of   a,  /9 
and  0. 

[Hint.     Let  0  =  oc  +  /3  and  obtain  sin  (</>  +  6);  then  replace  0  by  it 
value,  a  +  /3.  ] 

7.  Express  cos  (a  +  /8  +  ^)   in  terms  of  sines  and  cosines  of  a,  /9 
and  e, 

8.  Reduce  the  combination  of  two  simple  harmonic  motions  5  cos  t  - 
12  sin  t  to  the  form  r  cos  (t  -\-  0). 

9.  Keduce  3  sin  i  +  4  cos  t  to  the  form  rsin  (t  -\-  0). 

10.   Reduce  each  of  the  following  to  the  product  of  a  number  and  th 
sine  or  the  cosine  of  a  single  angle  : 

(a)  sinx  — 2cosx.  (  e)   \/3  cos  x  —  sin  x, 

(6)  3  cos  2/  —  4  sin  y.  (/ )  sin  ?/  +  .5  cos  y. 

(c)  5  cos  ^  +  12  sin  0,  (  g)  .7  cos  0  —  sin  0. 

(d)  3  sin  i  —  3  cos  t.  (h)  .55667  sin  c  +   5  cos  c. 


XI,  §  75]  ADDITION  FORMULAS  101 

11.  Given  two  forces  of  intensities  2  and  3  that  make  angles  of  30® 
and  120°,  respectively,  with  the  positive  x-axLs  ;  find  the  horizontal  and 
the  vertical  components  of  their  resultant  without  finding  the  resultant 
itself  ;  find  the  same  quantities  by  using  the  resultant. 

12.  Given  .66  sin  c  +  .5  cos  c  =  —  .34,  find  an  angle  ^,  and  a  number  r, 
such  that  .56  sin  c  +  .6  cos  c  =  r  sin  (c  +  ^),  by  means  of  §  70.  Then, 
from  r  sin  (c  +  ^)  =  —  .34,  find  sin  (c  +  ^),  and  therefore  (from  the 
Tables)  find  c  -\-  $.     Hence  find  c. 

74.  Double  Angles.  Since  formulas  (6)  and  (7),  §  71,  are 
true  for  all  angles,  they  hold  when  a  =  a,  any  angle  whatever, 
and  /3  =  a,  the  same  angle  ;  hence, 

sin  (^a-\-a)=  sin  a  cos  a  -\-  cos  a  sin  a, 
and  cos  (a  -\-  a)=  cos  a  cos  a  —  sin  a  sin  a. 

Therefore  the  following  formulas  hold  for  any  angle  what- 
ever : 

(10)  sin  2  a  =  2  sin  a  cos  a ; 

(11)  cos  2  a  =  cos2  a  —  sin^  a ; 
or,  since    sin^  a  +  cos^  a  =  1, 

(12)  cos  2  a  =  1  —  2  sin^  a  =  2  cos2  a  —  1. 

75.  Tangent  of  a  Sum  or  of  a  Difference.  Since  formulas 
(6)  and  (7)  hold  for  all  values  of  a  and  ^,  the  formula 

sin  (a  +  )S)  __  sin  a  cos  fi  +  cos  a  sin  fi 
cos  (a  +  )8)  ~"  cos  a  cos  fi—aina  sin  fi 

holds  good  for  all  values  of  a  and  ^  except  those  which  make 
cos  (a-j-/S)=0,  i,e.  except  when  a-}- ft  =  90°,  or  270°,  or  an 
angle  that  differs  from  one  of  these  by  an  integral  number  of 
times  360°.  For  example,  it  does  not  hold  for  a  =  47**,  P  =  43^ 
Dividing  both  numerator  and  denominator  by  cos  a  cos  p,  we 
obtain  the  formula 

(13)  tan(a+P)=  tana-ftanP 
^     ^  V    Try     i_tanatanp 

which  holds  for  all  angles  a  and  ^  such  that  a,  )S,  and  «  4-  ^ 
have  tangents. 


102  PLANE  TRIGONOMETRY  [XI,  §  75 

Similarly  from  formulas  (8)  and  (9),  we  obtain 

(14)  tan(a-P)=i^?^LIl^, 

which  holds  for  all  angles  a  and  j8  such  that  a,  p,  and  a  —  p 
have  tangents. 

From  formulas  (10)  and  (11)  we  find 

,^^.  X      ft  2  tan  a 

(15)  tan  2a  = , 

^  ^  l-tan2a 

which  holds  for  every  angle  a  such  that  a  and  2  a  have  tan- 
gents. The  same  formula  ma}^  be  obtained  directly  from  (13) 
by  putting  a  in  place  of  p. 

76.  Applications.  The  formulas  of  this  chapter  are  fre- 
quently used  for  reducing  expressions  whose  values  are  to 
be  calculated,  to  a  form  in  which  logarithms  may  be  used. 

Example.     Suppose  the  height  of  an  object  CD  is  to  be  determined 
and   that  it  is  not   convenient  to  measure  a  base  hne  bearing  directly 
toward  the  base  G.     The  following  method  is  then 
sometimes  employed.     The  angle  of  elevation  a  is 
measured  from  some  convenient  point  A\  a  line 
AB  =  d  is  then  measured  at  right  angles  to  the 
line  A  G  ;  finally  the  angle  of  elevation,  j3,  is  ob- 
served from  B.     The  height  h  can  then  be  de- 
termined by  solving   a  succession   of   triangles. 
With  the  aid  of  the  formulas  of  this  chapter  it 
Fig.  86.   ^^B       i^  frequently  possible  in  such  cases  to  reduce  the 
calculation  to  a  single  logarithmic  computation. 
In  the  case  just  mentioned  we  have 

BG  =  hctn^  AG  =  h  ctn  a, 

d2  =  BG^  _  Jc^  =  7^2  (ctn2  ^  -  ctn2  a) 
=  h^  (ctn  )3  —  ctn  a)  (ctn  /3  +  ctn  a) 
__  ,  2  (sin  a  cos  j3  —  cos  a  sin  /3)  (sin  a  cos  j8  +  cos  a  sin  /3)  ^ 
~  sin2  a  sin2  /3  ' 

hence,  using  formulas  (6)  and  (8),  we  have 
,  _■  dsinasiujS 

Vsin  (a  —  /3)  sin  (a+^1 
Let  the  student  show,  by  opening  a  book  and  studying  the  dihedral 
angle  formed  by  two  leaves,  that  a  >  )3. 


XI,  §  76]  ADDITION  FORMULAS  103 

EXERCISES    XXVn.  — SECONDARY    FORMULAS  —  APPLICATIONS 

1.  Find  sin  15°,  cos  15°,  tan  15°  from  the  known  values  of  sin  30^, 
cos  30°,  tan  30°,  and  sin  45^,  cos  45°,  tan  45°.     [Hint.     15°  =  46°  —  30°.] 

2.  Find  tan  75°,  tan  105°,  sin  165°,  cos 255°.     [Hint.   7 5°= 45'= -|- 30°.] 

3.  Given  sin36°52' =  .0 ;  find  the  sine,  cosine,  and  tangent  of 
66°  52';  find  sin  73°  44'. 

4.  Given  tan  26°  34'  =  .5  ;  find  sine,  cosine,  tangent  of  71°  34';  find 
tan  53°  8'. 

6.  Given  sin  a  =  5/13  and  90°  <  a  <  180°  ;  cos  /3  =  8/17  and  0°  <  /3  < 
90°;  find  sin  (a  —  /3) ,  cos  (a  —  /3),  tan  (a  +  /3),  sin  2  a,  cos  2  /3. 

6.  Given  tan  a  =  15/8  and  0°  <  a  <  90°  ;  cos  /3  =  4/5  and  270°  <  /S  < 
360°;  find  sin  (a  -  )3) ,  cos  (/3  —  a) ,  tan  2  a,  cos  2  j8. 

7.  Given  sin  ex.  =  1/3  and  90°  <  a  <  180^ ;  find  sin  (135°  -  «)  and 
tan  2  a. 

8.  The  angular  elevation  of  an  object  from  an  upper  window  is  ob- 
served to  be  a.  The  angular  elevation  from  a  point  on  the  ground  h  feet 
directly  beneath  the  window  is  /3.  Show  that  the  height  of  the  object  is 
h  sin  /3  cos  a  -T-  sin  (/3  —  a). 

9.  To  determine  the  difference  in  elevation  of  two  stations,  a  flagstaff 
of  known  height  h  is  held  at  the  upper  of  two  stations  and  the  angles 
of  elevation  of  its  top  and  bottom  are  observed  to  be  a  and  j8,  respectively. 
Show  that  the  difference  in  elevation  of  the  two  stations  is  h  tan  /3  —  (tan 
a  —  tan  j8) ;  reduce  this  expression  to  a  form  convenient  for  logarithmic 
computation. 

10.  A  tree  leans  directly  toward  two  points  of  observation  distant  a 
and  &,  respectively,  from  its  foot.  The  angles  of  elevation  of  the  top  of 
the  tree  from  these  two  points  are  a  and  /3.  Show  that  the  perpendicular 
height  of  the  tree  is  (6  —  a)  -r-  (cot  /S  —  cot  a)  ;  reduce  this  expression  to 
a  form  suitable  for  logarithmic  computation. 

11.  Prove  that  sin  Sa  =  sin  a  (3  -  4 sin^  «)  =  sin  a  (4  cos2 a  -  1),  and 
state  for  what  values  of  a  it  holds.     Use  formuUis  (6)  and  (7). 

12.  Prove  that  cos  3  a  =  cos  a  (4  cos2  a  —  3)  =  cos  «  (1  —  4  sin*  «),  and 
state  for  what  values  of  a  it  holds.     Use  formulas  ((>)  and  (7). 

13.  Prove  that  tan  Sa=  ^  tan  «  -  tan^  a  ^^^^^  ^j^^^  ^y^^^  j^^  j^^^j^,^  f^^ 

1—3  tan2  a 
all  values  of  a  such  that  «  and  3  a  have  tangents. 

14.  Prove  that  sin  (45°  +  a)  sin  (45°  -  «)  =  1/2  cos  2  a  for  all  values 
of  a. 

15.  Prove  that  sin  («  +  /3)  sin  («  -  ^)=  sin2  «  -  sin2  ^  for  all  values 
of  a  and  /3. 

16.  Prove  that  cos  (a+/3)  cos  fi  +  sin  (a  +  /3)  sin  /3  =  cos  a. 


104  PLANE  TRIGONOMETRY  [XI,  §  77 

77.  Functions  of  Half  Angles.     The  formulas 

cos^  a  +  sin^  a  =  1 
and 

cos^  a  —  sin^  a  =  cos  2  a 

are  true  for  all  values  of  a.     If  we  subtract  one  of  these  from 
the  other,  and  if  we  also  add  them,  we  obtain  the  formulas : 

(16)  2sin2a=:l  — cos2a, 

(17)  2  cos2  a  =  1  +  cos  2  a. 

These  formulas  are  true  for  all  values  of  a ;    for  a  =  a' /2 

they  become 

2sin2(a72)=l-cosa' 
and 

2  cos2  (a72)  =  1  +  cos  a', 

or  since  these  are  true  for  all  values  of  a\  we  may  write 

(18)  sin  (a/2)  =±'^- 


cos  a 


(19)  .        cos(a/2)=±V^^4^, 

which  hold  good  for  all  values  of  a.     The  same  formulas  may 
be   obtained   from     (12)    by    solving    for    sin    (a'/ 2),   or   for 
cos  (cc'/2),  after  putting  a^/2  for  a. 
Erom  (18)  and  (19)  we  get  by  division 


ir^ 


cos  a  sm  a 


(20)       tan  a/2  =  ±  \/  t— =  z 

^  1  +  cos  a      1  +  cos  a 


sm  a 


which  hold  for  all  values  of  a  except  when  a  denominator 
vanishes.  The  ambiguity  of  sign  of  the  radical  is  determined 
in  a  given  case  by  the  fact  that  tan  (ct/2)  is  positive  or  nega- 
tive according  as  a/2  is  or  is  not  in  the  first  or  second 
quadrant. 

The  relations  between  an  angle  and  its  half  are  frequently 
useful  in  problems  that  relate  to  a  chord  of  a  circle  and  the 
angle  which  it  subtends  at  the  center ;  this  occurs,  for  example 


XI,  §  77] 


ADDITION  FORMULAS 


105 


in  laying  out  railroad  curves  where  it  is  convenient  to  make 
measurements  along  chords  of  the  curve.  This  is  illustrated 
in  some  of  the  exercises  below.  The  relations  are  also  useful 
in  simplifying  trigonometric  expressions  and  in  adapting  for- 
mulas to  logarithmic  computation. 


EXERCISES   XXVIII.— -HALF- ANGLE   FORMULAS 

1.  Find  the   sine,  the  cosine,  and  the   tangent  of  22°  30'  from  the 
kfiown  values  of  sin  45°,  cos  45°,  tan  45°. 

2.  Find  the  sine,  cosine,  and  tangent  of  15°. 

3.  Given  that  sin  a  =  4/5,  and  that  a  is  an  acute  angle  ;  find  sin  (a/2) 
and  tan  (a/2). 

4.  Given  tan 26°  34'=  1/2  ;  find  tan  13°  17'. 

5.  Given  tan  36°  52'  =  3/4  ;  find  sine,  cosine,  and  tangent  of  18°  26'. 

6.  If  r  denotes  the  radius  of  the  circle  in  the  accom- 
panying figure,  c  a  chord,  and  6  the  angle  which  c  sub- 
tends at  the  center  ;  show  that  sin  (6/2)  =c/(2  r) . 

7.  In  the  figure,  draw  the  line  BD  tangent  to  the 
circle,  and  AD  perpendicular  to  BD  from  the  opposite 
end  of  the  chord  BA.  Show  that  (a)  ZABD  =  6/2  ; 
(6)  BD  =  AB  cos  (6/2)  ==  2  r  sin  (6/2)  cos  (6/2)  = 
r  sin  6. 

8.  Prove  that  tan  (45°  -f  a/2)  =  sec  a  +  tan  a,  if  tan  a  exists. 

9.  Prove    that    tan  (45°  +  a/2)  tan  (45°  —  a/2)  =  tan  45°   if    tan  a 
exists. 

10.  Prove  that  tan  (a/2)  +  2  sin2  (a/2)  ctn  a  =  sin  a,  ilsma=^  0. 

11.  Prove  that  tan  (a/2)  +  ctn  (a/2)  =  2  esc  a,  if  sin  a  =7^  0. 

12.  Prove  that  [sin  (a/2)+  cos  (a/2)]2=  1  +  sin  a  for  all  values  of  a. 

13.  Prove  that  [sin  (a/2)  —  cos  (a/2)]2  = 
1  —  sin  a  for  all  values  of  a. 

14.  In  the  figure,  COA  is  a  diameter  of  a  circle 
of  radius  r  ;  A  OP  =  a  is  any  acute  angle  ;  OCP  = 
a/2,  by  geometry  ;  and  PB  is  perpendicular  to 
OA.     Show  that 

'  OB  =  r  cos  a,    BP  =  r  sin  a,    BA  = 

r  vers  a,    CB  =  r(l  +  cos  a). 


Fig.  87. 


CP  =  ^PB^  +  CB^  =  rV2(l  +  cos  a). 


Fig. 


106  PLANE  TRIGONOMETRY  [XI,  §  78 

15.    From  Ex.  14,  show  that  the  functions  of  a/2  can  be  read  directly 
from  the  figure  in  the  form  : 

sin  (a/2)  =  ^"^^^         =  Jl-cosa  . 

rV2(l  +  cosa)       ^         2 

cos  (a/2)  =      ^  +  ^o^<^      =    11  + cos  a . 
V2(l  +  cos  a)       ^        2 

Vl  —  cos2  a        / 1  —  cos  a     1  —  cos  a 


+      /^/oN         sin  a  \/l  — cos2a     ^/l 

tan  (a/2)  = = =  \  ~ 

1  +  cos  a        1  +  cos  a         ^  1 


1  +  cos  a        1  +  cos  a         ^  1  +  cos  a         sin  a 

16.  If  a  numerical  value  of  any  function  of  a  is  given,  all  the  other 
functions  of  a  and  of  a/2  can  be  found  geometrically  from  Ex.  14.  Thus, 
if  sin  a  =  4/5  is  given,  lay  off  0P=  5,  BP=4:;  then  OB  =  \/52  -  42=  3. 

Hence,       CB  =  S,  BA=z2;  and  CP  =  ■y/'cB'^  +  BP^  =  VS^  +  42=  \/80. 
It  follows  that 

sin  a  =  4/5,  cos  a  =  3/5,  tan  a  =  4/3, 

sin  (a/2)  =  4/V80  =l/\/5  ==  \/5/5, 
cos  (a/2)=  8/\/80  =  2/V5  =  2V5/5, 
tan  (a/2)  =  4/8  =  1/2. 

17.  Eind  the  remaining  functions  of  a  and  those  of  a/2  by  means  of 
Ex.  16,  if  cos  a  =  5/13  ;  if  tan  a  =  1/3. 

18.  The  remaining  functions  of  (a/2)  and  those  of  a  can  be*  found 
when  any  function  of  a/2  is  given  from  the  figure  of  Ex.  14,  by  dropping 
a  perpendicular  from  O  to  CP.     Do  this  if  tan  (a/2)  =  3/4. 

19.  Since,  in  the  figure  of  Ex.  14,  by  geometiy  BP^  =  CB  •  BA,  show 
that  (1  +  cos  a)  vers  a  =  sin2  a. 

20.  Derive  trigonometric  formulas  from  the  geometric  identities 
(Ex.  14)  :  __  

BP'PA  =  AB\  BP'  CP=  CB^. 

78.  Factor  Formulas.  In  adapting  trigonometric  formulas 
to  logarithmic  computation  it  is  often  desirable  to  express  the 
sum  (or  difference)  of  two  sines  (or  cosines)  as  the  product  of 
other  functions. 

Example  1.     Reduce  sin  35°  -f  sin  15°  to  the  form  2  sin  25°  cos  10°. 

To  do  this,  set         x-\-y  =  35°,  x  —  y  =  15°, 

and  solve  for  x  and  y  :        x  =  25°,  y  =  10°. 
Then  sin  (x  -\-  y)=  sin  x  cos  ?/  +  cos  x  sin  y, 

sin  (x  —  y)=sinx  cos  y  —  cosx  sin  y  ; 
whence,  adding,     sin  (x -\- y)  +  sin  (x  -  y)  =  2  sin  xcosy  ; 
substituting  x  =  25°,  y  =  10°,  we  get  sin  35°  H-  sin  15°  =  2  sin  25°  cos  10°. 


XI,  §  78]  ADDITION  FORMULAS  107 

Example  2.     Reduce  sin  s  —  sin  (s  —  c)  to  a  product, 
where  s  =  (a  +  6  +  c)/2. 

Let  x-{-y=s,  x  —  y  =  s  —  c;  then  aj  =  (a  -f  6)/2, 2/  =  c/2, 
and  sin  (x  +  ?/)  =  sin  x  cos  2/  +  cos  aj  sin  y^ 

sin  («  —  2/)  =  sin  aj  cos  y  —  cos  x  sin  ?/ ; 
subtracting  sin  (x  -\-  y)  —  sin  (x  —  2/)  =  2  cos  x  sin  2/, 

whence     sin  s  —  sin  (s  —  c)  =  2  cos  [(a  +  6)/2]  sin  (c/2). 

EXERCISES  XXIX.— FACTORING 

1.    Reduce  each  of  the  following  forms  to  products  : 

(a)  sin  70°  -  sin  10°.  (6)  sin  70^  +  sin  50°. 

(c)  sin  13°  +  sin  41°.  (d)  sin  34°  -  sin  19°. 

(e)  cos26°--cos35°.  (/)  sin  43°  +  sin  28°. 

(g)  cos 20°  +  cos  10°.  (h)  cos 61°  -  sin  11°. 

.. .   sin  15°  +  cos  45°  , . .  sin  28°  +  sin  12° 


cos45°  — sinl5°  ^       cos  28°  +  cos  12° 

(k^  sii^  ^4°  +  sin  16°  m  ®^^  ^^°  ~  ^^^  ^^° 


sin  64°  -  sin  16°  cos  40°  —  cos  80° 

2.  Prove  that  cos  (x  +  2/)  +  cos  (x  —  y)=2  cos  x  cos  y, 

3.  Prove  that  cos  (x  +  2/)  —  cos  (x  —  2/)  =  —  2  sin  x  sin  y. 

4.  Prove  that 

cos  A  -\-cosB  =  2  cos ^  "^  ^ cos^""^. 
2  2 

by  substituting  ^  =  x  +  ?/,  -B  =  x  —  y  in  Ex.  2. 

5.  Prove  by  means  of  Ex.  3  that 

A  4-  B        A 
cos  i4  —  cos  5  =  —  2  sin     ^     sin  — 


2  2 

6.  By  the  method  of  Example  1,  §  78,  show  that ' 

sin  i4  +  sin  5  =  2  sin  ^-^  cos  ^^—-? . 
2  2 

7.  By  the  method  of  Example  2,  §  78,  show  that 
sin  i4  -  sin  B  =  2  cos  ^^-^  sin  ^^-^. 


2 

=  lan — —^^iiL 
sin  X  —  sin  2/ 


8.   Prove  ?iH^±.!iM  =  tan^+^ctn 


9.    Prove  cosx+cos2/^_  ctn ^±J^ ctn ^^li^. 
cos  X  —  COS  2/  2  2 

10.  Prove  ^'"g  +  sin^O  ^  ^^^  ^^/g). 
cose  — cos2« 


108 


PLANE  TRIGONOMETRY 


[XI,  §  78 


11.  Prove  si"(2x-3j/)+sin3y  ^  ^^^  ^ 

COS  (2  X  —  3  y)  +  COS  3  2/ 

12.  sin  (45''  +  x)  +  sin  (45°  —  x)  =  V2  cos  x. 

13.  sin  3  X  +  sin  6  X  =  2  sin  4  x  cos  x. 

14.  If  a  +  6  4-  c  =  2  s,  show  that 

(a)  cos  (6  —  c)—  cos  a  =  2  sin  (s  —  h)  sin  (s 
(6)  cos  a  —  cos  (6  +  c)  =  2  sin  s  sin  (s  —  a)  ; 


-c): 


(c) 


sin  (8  ■ 


c)  _       tan  I  c 
tan  J  (ct  +  6) 


15. 


16. 


sin  s  +  sin  (s  —  c) 

tan  X  tan  y    __  sin  x  sin  y 
tan  X  —  tan  y      sin  (x  —  2/) 

The  so-called  "method  of  offsets"  for  laying 
out  a  circular  track  is  illustrated  in  the  adjoining  figure. 
The  track  OAB  is  tangent  at  0  to  05',  and  the  dis- 
tances OA',  A'B'^  A' A,  CB,  are  easily  shown  to  be 
as  marked  in  the  figure,  where  a/2  =  ZAOA'  is  half 
the  angle  at  the  center  subtended  by  a  100-foot  chord. 
In  practice,  the  hne  OA'B'  is  run,  and  A'  and  B' 
marked.  Show  that  B'B,  the  distance  actually  to  be 
laid  off  from  B',  is 

B'B  =  A' A  +  CB  =  200  sin  a  cos  (a/2).' 


8lnC3«/2), 


CHAPTER   XII 


GRAPHS   OF   TRIGONOMETRIC   FUNCTIONS 

79.  Scales  and  Units.  The  graph  of  the  function  sin  a;  is  a 
curve  passing  through  all  points  whose  coordinates  (x,  y), 
satisfy  the  equation  y  =  sin  x.  The  graph  of  any  other  trigo- 
nometric function  as  cos  a;,  tan  ic,  etc.,  is  similarly  determined. 

The  radian  is  the  unit  angle  commonly  used  in  plotting  the 
graphs  and  in  the  further  study  of  the  trigonometric  functions 
in  the  Calculus  and  in  other  advanced  mathematical  subjects. 
Unless  otherwise  specified,  the  equation  y  =  sin  x  is  understood 
to  mean  that  y  is  the  sine  of  x  radians  ^  as  explained  in  ^  64. 

In  plotting  curves  it  is  of  advantage  in  many  ways  to  make 
the  horizontal  and  vertical  scale  units  the  same,  and  this 
should  be  done  if  not  too  inconvenient,  f 

80.  Plotting  Points.  In  Table  V  are  given  the  values  of  the 
sine,  cosine,  and  tangent  of  acute  angles  measured  in  radians 
which  are  very  convenient  for  plotting  the  graphs  of  these 
functions  on  cross-section  paper. 

81.  Graph  of  sinjc.  Draw  a  pair  of  coordinate  axes  and 
choose  the  scale  unit  =  10  small  divisions  of  the  cross-section 
paper.  Take  from  Table  V  the  sines  of  the  angles  in  the  first 
quadrant  for  each  tenth  radian  and  tabulate : 


X 

0 

.1 

.2 

.3 

.4 

.5 

etc.     .     .     . 

1.6 

1.57 

?/  =  sin  X 

0 

.099 

.198 

.295 

.389 

.479 

etc.     .     .     . 

1.000 

*  In  any  case,  y  =  sin  x  means  that  y  is  the  sine  of  x  units  of  angle.  The 
right  angle,  the  60°  angle,  the  45"  angle,  the  degree,  or  any  other  angle  might 
be  chosen  as  the  unit,  if  it  were  convenient. 

't  If  we  were  to  take  the  two  scale  units  the  same  in  plotting  the  curve  y  = 
sin  X  where  the  unit  angle  is  the  degree,  one  arch  of  the  curve  would  be  180 
units  long  and  only  1  unit  high. 

109 


110 


PLANE  TRIGONOMETRY 


p:ii,§8i 


Plot  these  points  and  draw  a  smooth  curve  through  them  as 
OA  in  the  figure. 


|y     1     III      ill      1      II      II        m^^^^^^ 

/k                                -                    -  - 

-Jfr----""-^ 

s£              -        ^^ 

^S^               -                     s             -                                           -                   -      - 

^^                              _   ___i^_    

---  fi ^5" :  ■ 

i^                                                     s 

'B^  '        :        i                      '^  ^^                 ::^:              -S^? 

£— :: — -+ ^^^ ± F-4^ 5-1^- 

9::::::::v::::t::±::::::::i|lv:::::±::::5fe$::::::::-,z5: 

5^      ^                        ^^                    ^? 

'^ _.^t 

->-- -^"■"r 

-        ^^        -     -              -              ,^ 

s                                     / 

^  !»  IS                                         ,*^- 

-                        -------                  -- -__.----^-. -_-        -        - 

Fig.  90. 

It  is  readily  seen  by  the  principles  of  §  68  that  the  exten- 
sion of  the  curve  through  the  second,  third,  and  fourth  quad- 
rants is  as  shown  by  AB,  BCj  and  CD ;  and  that  the  curve 
extends  to  the  left  and  to  the  right  of  the  origin  in  a  succes- 
sion of  arches  such  as  OAB,  BCD,  etc. 

The  graph  of  sin  x  can  be  drawn  without  the  aid  of  Table  V 
as  follows :  Choose  a  convenient  scale  unit  and  lay  off  on  the 

aj-axis  OP  =  -  =  1.57  approximately,  and  divide  this  segment 
into  a  convenient  number  of  equal  parts,  15  say ;  the  points  of 

r>  o 

division  correspond  to  a:=  0,  ^,   -^,  -^,  •••,  ^'     Take  from 

«jU     oyj      o\j  Z 

a  table  of  sines,  such  as  the  one  printed  on  p.  21  for  example, 

the  sines  of  the  angles  in  the  first  quadrant  for  each  6°  and 

tabulate : 


X 

0 

TT 

30 

2^ 
30 

37r 
30 

47r 
30 

etc.     .     .     . 

IT 

2 

y  =  sin  « 

0 

.105 

.208 

.309 

.407 

etc.     .     .     . 

1.000 

Plot  these  points  and  draw  a  smooth  curve  through  them. 


XII,  §  82] 


TRIGONOMETRIC  GRAPHS 


111 


Fig.  91. 


The  same  methods  may  be  used,  with  obvious  modifications, 
to  plot  the  graphs  of  cos  x^  tan  x^  and  in  fact  any  one  of  the 
trigonometric  functions. 

82.   Mechanical  Construction  of  the  Graph.     If  an  angle 

of  X  radians  be  laid  off  at  the  center  of  a  unit  circle  (i.e.  a 

circle  whose  radius  is  the  scale 

unit),  as  AOB  in  Tig.  91,  the 

numerical  measure  of   the  arc 

AB  is  the  number  of  radians  in 

the  angle,  Le.  x\    the  measure 

of  CB  'v.   sin  a;,  the  measure  of 

AD  is  tan  x^  the  measure  of  00 

is   cos  a?,  and   the  measure  of 

OB  is  sec  X, 
These  facts   can  be  used  to 

construct  the  graphs  of  these 

functions  without  the  use  of  any  tables  whatever.     If  we  lay 

off  on  the  ic-axis  a  segment  equal  in  length  to  the  arc  AB  and 

at  its  end  point  erect  a  perpendicular   equal  to  CjS,  its  end 

point  will  lie  on  the  graph  of  sin  x.     It  remains  to  show  how 

to  lay  off  a  line  segment  approximately  equal  in  length  to  a 

circular  arc.  If  the  arc  AB  is  a 
known  part  of  the  quadrant  AQ 
whose  measure  is  1.5708",  the  meas- 
ure of  ^jB  can  be  computed  and 
laid  off  with  a  scale.  This  will  be 
the  case  if  B  is  one  of  the  points 
of  division  which  divide  the  quad- 
rant into  a  number  of  equal  arcs. 
But  even  if  the  ratio  of  AB  to 
^Q  is  unknowii,  provided  AB<iAQ, 

it  can  be  approximately  rectified  as  follows. 

With  Q  as  center,  and  the  diameter  Qi?  as  radius,  strike  an 

arc  cutting  AO  produced  in  8\  draw  SB  cutting  AB  in  P. 


Fig.  92. 


112 


PLANE  TRIGONOMETRY 


p:n,  §  82 


Then  the  length  of  A*F  is  approximately  equal  to  the  length 
of  the  arc  AB.* 

To  use  this  method  for  constructing  the  graphs  of  sin  a?, 
cos  Xy  etc.,  draw  the  unit  circle,  as  in  Fig,  93,  tangent  to  the 
y-axis  at  the  origin  and  divide  the  radian  arc  into  a  convenient 
number  of  equal  parts,  say  5,  by  lines  from  S  to  the  points  .2, 
.4,  .6,  etc.,  on  the  ^/-axis  (the  last  division  of  the  quadrant  will 
of  course  be  only  .17+  long).     Mark  the  points  0,  .2,  .4,  .6,  .8, 


^Y 

y 

/ 

. 

y^ 

^ 

^ 

- 

y 

■^ 

^ 

"^ 

^ 

<-" 

^ 

'' 

S>| 

^ 

^ 

^ 

'■*v 

^X 

k 

»^ 

Kd-* 

"^ 

A 

^ 

L^ 

.8 

^ 

-^ 

^ 

s 

i^ 

s> 

yf- 

^ 

•'^. 

/ 

*■ 

N 

s 

^ 

y 

/^ 

•\\ 

/ 

s 

\ 

4 

u 

^^ 

/ 

\ 

\ 

X 

\ 

c 

0 

2  .< 

\  .( 

)  .i 

5 

. 

~s 

V 

~^ 

>^ 

\ 

v 

y 

N 

\ 

\ 

/ 

:os 

[s 

c 

V 

k. 

\ 

/ 

s 

:\ 

==- 

R 

. 

y 

•v 

^ 

^ 

"€ 

Fia.  93. 

1,  1.2,  1.4,  1.57,  on  the  oj-axis  and  erect  perpendiculars  equal 
to  the  ordinates  of  the  corresponding  points  on  the  arc.  These 
give  points  on  the  graph  of  sin  x. 

By  erecting  perpendiculars  to  the  oj-axis  equal  to  the  hori- 
zontal distances  from  CQ  of  the  corresponding  points  on  the 
arc  we  shall  get  points  on  the  graph  of  cos  x. 

By  drawing  radiating  lines  from  the  center  G  of  the  unit 

circle  through  the  points  of  division  of  the  arc  we  can  lay  off 

the  tangents  of  these  arcs  on  the  y-axis  and  construct  the  graph 

•# 

*  The  proof  of  this  cannot  be  given  until  the  student  has  studied  Calculus. 
The  distance  ^P  is  greater  than  x,  but  the  error  is  less  than  S)Ylx^.  The 
greatest  error,  about  .017,  occurs  when  AB  is  an  arc  of  about  74°  29',  or 
when  X  —  l.S^**^  approximately.  The  error  for  a  45°  arc  is  .007  and  for  a  quad- 
rant, .006. 


XII,  §  82]  TRIGONOMETRIC  GRAPHS  113 

of  tan  X ;  and  in  an  obvious  manner  (see  Fig.  91)  the  graph  of 
sec  a;  can  be  drawn.  These  graphs  can  be  extended  through 
the  other  three  quadrants,  and  to  the  left  of  the  y-Sixis,  as  in 
§  81.  If  the  angle  increases  beyond  2  tt  (radians)  the  values 
of  all  the  trigonometric  functions  repeat  themselves  and  the 
graph  from  x=2 tt  to  x  =  Air  will  be  a  repetition  of  those 
from  a;  =  0  to  oj  =  2  TT. 

Functions  which  repeat  themselves  as  x  increases  are  called 
periodic  functions.  The  period  is  the  smallest  amount  of 
increase  in  x  which  produces  the  repetition  of  the  value  of  the 
function.  Thus,  sin  a?  is  a  periodic  function  with  a  period  of 
2  TT,  while  the  period  of  tan  x  is  tt. 

EXERCISES    XXX.  — GRAPHS     OF    TRIGONOMETRIC    FUNCTIONS 

1.  Plot  the  graphs  of  the  following  functions  using  Table  V,  and 
Table  VI  when  necessary. 

(a)  cosx  (6)  tana;  (c)  versa 

(d)  ctnx  (e)  sec  a;  (/)  cscx 

(gr)  sin2x  (h)  cos2x  (i)   Vsinx 

2.  Plot  the  graphs  of  the  following  functions  without  the  use  of 
tables:     (a)  cosx  (6)  tanaj  (c)  secx 

3 .  Plot  the  graph  of  cos  x  by  dividing  the  second  quadrant  of  the 
unit  circle  into  fifths  of  a  radian  (see  Fig.  91)  and  making  use  of  the 
fact  that  cos  x  =  sin  (7r/2  +  x) . 

4.  Plot  on  the  same  axes  the  graphs  of  sinx,  sinjx,  sin2x,  and 

2  sin  X. 

5.  Plot  on  the  same  axes  the  graphs  of   cosx,  cosjx,  cos3x,  and 

3  cos  X. 

6.  Discuss  the  graphs  of  sin  x/n,  sin  nx,  and  nsinx  (where  n  Is  a 
natural  number)  in  view  of  the  results  of  Ex.  4  and  6. 

7.  Plot  the  graph  of  sin  x  +  cos  x  by  adding  the  corresponding  ordi- 
nates  of  the  curves  y  =  sin  x  and  y  =  cos  x  plotted  on  the  same  axes. 

8.  Plot  the  graphs  of  the  following  functions  by  adding  ordinates  : 

(a)  sin  X  —  cos  X  (  6  )  2  sin  x  +  cos  x 

(c)  tan X  — 2 sin X  (d)   —  cosx  (i.e.  0  —  cosx) 

(e)  x  +  sinx  (/)  x  — cosx 

9.  Plot  on  the  same  axes  the  graphs  of  sin  x,  and  sin  (x  —  ^/6). 

10.   Plot  on  the  same  axes  the  graphs  of  sin  x,  cos  x,  and  cos  (x  —  Tr/2), 
I 


114  PLANE  TRIGONOMETRY  [XII,  §  84 

83.  Inverse  Functions.  We  have  seen  in  §  69  that  the 
equation 

(1)  y  =  sin  X 

can  be  solved  for  a;  if  2/  is  any  number  whatever  between  —  1 
and  +  1,  and  that  there  are  an  infinite  number  of  solutions. 
Any  one  of  these  solutions  is  denoted  by  ^ 

(2)  X  =  arcsin  y. 

If  we  suppose  that  the  angle  is  measured  in  radians,  (2)  means 
that  X  is  the  number  of  radians  in  an  angle  (or  arc)  whose  sine 
is  y;  it  is  read  "  arc  sine  y  "  or  "  an  angle  whose  sine  is  3/." 

Likewise  arccos  y  denotes  an  angle  whose  cosine  is  y ;  arc- 
tan  y  denotes  an  angle  ivhose  tangent  is  y. 

The  expressions  y  =  sin  x,  x  =  arcsin  y,  are  two  aspects  of 
one  relation,  just  as  are  the  two  statements  "  A  is  the  uncle 
of  B  "  and  "  B  is  the  nephew  of  A  " ;  either  one  implies  the 
other ;  both  mean  the  same  thing. 

As  we  wish  to  study  the  arcsine  function,  and  in  particular  to 
compare  it  with  the  sine  function,  it  is  convenient  and  customary 
to  think  of  it  as  depending  on  the  same  variable  x,  and  write 

(3)  y=  arcsin  x,     [i.e.  x  =  sin  2/]. 

We  note  that  (3)  is  obtained  from  (1)  by  two  steps,  (a)  solv- 
ing (1)  for  x;  and  (b)  interchanging  x  and  y  in  (2).  Two  func- 
tions so  related  that  each  can  be  obtained  from  the  other  in 
this  manner  are  called  inverse  functions ;  each  is  the  inverse 
of  the  other. 

In  the  same  sense,  y  =  cos  x  and  y  =  arccos  x;  y=z  tan x  and 
2/=arctan  x-,  y=  sec  x  and  y  =  arcsec  x-,  y  =  vers  x  and  y  = 
arcvers  x ;  etc.,  are  inverse  functions. 

84.  Graphical  Representation  of  Inverse  Functions.  Since 
the  equations 

(1)  y  =  sm  X    and     (2)     x  =  arcsin  y 

*  The  notation  sin-i  y  also  is  used  very  frequently  to  denote  arcsin  y,  it  is 
necessary  to  notice  carefully  that  sin— 1  y  does  not  mean  (sin  y)-'^. 


XII,  §  84] 


INVERSE  FUNCTIONS 


115 


are  equivalent,  the  same  pairs  of   values   of   x  and  y  which 
satisfy  one  of  them  satisfy  the  other.     Hence  either  of  these 


[:::±t-iT-r-i:T-4T:^d±:±::::::i:i^ 

11                   -III        lijjjJjJjJ  11  iJ  jJjjJjJ  lJJ4J44+y^--^      1 

IHTtTT:      ffUl             y  s  n-xum 

S~         ""      i                          -         --            T      ±4-                 4--          --3-^         -             --i      .      .      .      S^JJ         .      _      . 

::5-::^::::::T:::::g:irr:J::rrr,:::;!^::::::::ffi::::::^:::::::::: 

^,±4.. ;r5f^_.    .     ..4./^...[ y^\ __::i;|::?v' -"==  =  =  ■ 

t^^-" ^ T-47--T-r"--ti^4--!j — i.j-i--.t>^ a.-(i-_.^ -X- 

^                      ^s                                 '        1                   1       /       1                                                    '                        -         -  -            >.         " 

s                          'ML/                                                       '                                                       ^v. 

J                      '       1           1    >                                                          '                     "      "  1    ^s" 

^  * ;     1       >  T    .      ,1                   . . • . .    .  _         _  i.   .    . . . . 

rv^^__4;._  J,«£   1      ._U^         j_^ ^_; _:__. 

=.Jc-=?=;,4,J.,Jj^.-.a,j,^-,^,Jo-L-i---LlJJ- -_Xt 

1       II       L  J           1         \  '  :  \  •  .  .  \     ■  \     \     \  \  \           1       i  !             1  !     !  II  1 

Fig.  94. 

two  equivalent  equations  is  represented  graphically  by  the 
curve  drawn  in  Fig.  94. 

From  the  manner  in  which  equation 

(3)  y  =  arcsin  x 

is  derived  from  (2)  it  follows  that  the  graph  of  arcsin  x  is 
obtained  from  the  graph  of  sin  x  by  interchanging  the  x-  and 
2/-axes;  or,  what  gives  the  same  result,  by  leaving  the  axes 
fixed  and  rotating  the  curve  through  an  angle  of  180°  about 
the  line  through  the  origin  which  makes  an  angle  of  45°  with 
the  cc-axis.     The  result  is  shown  in  Fig.  96,  p.  116. 


llllllllllllillllllfiiyi!i!»igi#»lil^ 

Fig.  95. 


116 


PLANE  TRIGONOMETRY 


pen,  §  84 


Similarly  from  the  graph  of  cos  x,  Eig.  95,  we  derive  the 
graph  of  arccos  x  in  Eig.  97 ;  and  in  the  same  way  the  graphs 
of  arctan  aj,  arcsec  a?,  arcctn  aj,  arccsc  x,  arcvers  x,  can  be  drawn 
from  those  of  tan  x,  sec  x,  ctn  x,  esc  x,  vers  x. 


:::::::  ::::::::::Tr :?:::-::::: 

±h 

::::::::::^-:-::S^-::::-:: 

:::::::::::::::::^:::::::::: 

C-0.-_WLd^R 

___X _/^ X 

llMlfi^JMU^ 

:::::::::^;^^:i::#::::T:::: 

/^ 1 i 

t     

__:^__::___::  +  ::     ::::::::::: 

i[ J— T— 1 

'i"T^T"l 

—  J. ^ 

:::^r:::::  :::=:===     :===:::= 

^ _ 

3^ 

1    hrhU    M    II 

:::i::::M:::::::::::::::::: 

ml"^h^    M n 

zr SJ— 

^ — 5ss"p!i 

._..___. ....^__.__ 

rnwHmmiffliy 

::  ::::::::::::::::::^F :::: :::: 

1                 1       |\  1     1 

::G;:::::::::::k;::  i^riin.x: 

M            0| 

:::::::::::: ::::::::2 ::::::::: 

:::  :::::::   :::::::2:::::::::: 

1         P'^fF#-^  II 

:::::::::H:;?I::::::::::::: 

-.=17/1^ 

:::::::::g::::::::::::::::: 

-:::-.^S--:::--:;:;:; 

Fig.  96. 


Fig.  97. 


EXERCISES  XXXI.  — INVERSE  FUNCTIONS 

1.  Draw  the  graph  of  y  =  arcsin  »  as  in  §  82. 

2.  Draw  the  graph  oiy  —  arccos  jc  as  in  §  82. 

3.  Draw  the  graph  oiy  —  arctan  x. 

4.  Draw  the  graph  oiy  =  arcsec  x. 


XII,  §  84] 


INVERSE  FUNCTIONS 


117 


Fig.  98. 


1        .                   '     M    pf 

:::::::::l:it:fe:z^x :::::::::: 

-- 

X_              ,*  \^ 

-- 

"":     ::::"  t"  /'*:::::::""  : —  : 

.,j:  4    .^ : 

'rr^    -    .  J :        : 

IJt 

_z^._.T _ ; 

'"      2L~r              "• 

-  :y-s  arx :  £  r : : : : : 

tT.         .!j __       ._    __ 

J=iC-' '■ 

T 

L+:  l-JY 4 

— 

" I'lX"  .-ii:  I""!""  :  "". 

j_.  —  ^. 

~&   — X"  :: :      : 

_  _  .  -_■»(_-    1^5]  p_:- 

;::::;:;;||^|g:;::;;;;:;;:; 

IJ^teN 

i: 

-*-j- _     _: 

zit 

i^' a  " " "  "           I 

m           1^^ 

::-;?^:::::^::::-^ 

I: 

■:!!!!"■;::::::: 

1     1  |i||i'  li'  Ml    IIIH^ 

-_\ 

\i\:\ll\-W^^^ 

Zl 

i ::  = 

-  =  =  " 

-■ 

._    .         .-    _^  .    ._._.-  ,,« 

T                          ^!-.     

•:::::::::::::::: 

rr  ~^  17 '  R" 

=; 

::::::::::±L:±::i: ::;,:::::::::: 

1 1  i  1 1 1 1 1 1 1 M  1 1 1 1 1 1 

i 

Fig.  99. 


LOGARITHMIC  AND 
TRIGONOMETRIC   TABLES 


LOGARITHMIC  AND 
TRIGONOMETRIC  TABLES 


REVISED   EDITION 


PREPARED  UNDER  THE  DIRECTION  OP 
EARLE  RAYMOND  HEDRICK 


NetD  gotfe 

THE  MACMILLAN  COMPANY 

1921 

All  rights  reserved 


Copyright,  1913  and  1920, 
By  the  MACMILLAN  COMPANY 


Set  up  and  electrotyped.      Revised  edition  published  August,  1920. 


J.  8.  Gushing  Co.  —  Berwick  &  Smith  Co. 
Norwood,  Mass.,  U.S.A. 


PREFACE 

The  present  edition  of  this  book  contains  several  tables  not  contained 
in  the  previous  editions.  The  probability  of  the  occurrence  of  errors  has 
been  minimized  by  using  electrotype  reproductions  of  the  tables  previ- 
ously included,  even  when  changes  were  made.  Remarkably  few  errors 
existed  in  the  original  edition  ;  what  few  have  been  discovered  have  been 
corrected. 

Minor  changes  only  occur  in  the  earlier  pages.  Care  has  been  taken 
to  preserve  the  page  numbers  of  the  principal  tables  up  to  page  114,  so 
that  older  editions  may  be  used  in  class-work  without  confusion,  and 
texts  which  contain  the  principal  tables  may  be  used  in  the  same  class. 

Among  the  minor  changes  are  the  insertion  of  a  condensed  table  of 
logarithms  and  antilogarithms  (Table  la,  p.  20) ,  the  insertion  of  a  table 
of  values  of  S  and  T  for  interpolation  in  logarithmic  trigonometric 
functions  (Table  Ilia,  p.  45),  and  the  insertion  on  pages  1-19  of  the 
logarithms  of  a  few  important  numbers  at  appropriate  points. 

The  principal  changes  follow  page  114.  Tables  VIII  and  IX  (pp.  115- 
122)  make  reasonably  complete  the  tables  of  hyperbolic  functions 
formerly  represented  only  by  Table  XII  (pp.  112-114):  These  functions 
are  of  increasing  importance,  notably  in  Electrical  Engineering-. 

The  table  of  haversines  (Table  X,  pp.  123-125)  will  be  welcomed 
particularly  by  those  interested  in  navigation. 

The  table  of  factors  of  composite  numbers  and  logarithms  of  primes 
(Table  XI,  pp.  126-127)  has  obvious  uses. 

Tables  XII  a,  6,  c,  df,  e,  /,  pages  128-132,  are  intended  for  work  in- 
volving compound  interest,  annuities,  depreciation,  etc.  They  will  be 
useful  for  statistics,  insurance,  accounting,  and  the  mathematics  of 
business. 

The  same  care  has  been  exercised  to  eliminate  errors  in  the  new  tables 
that  resulted  in  so  great  a  degree  of  reliability  in  the  original  edition  of 
these  tables. 

E.  R.  HEDRICK. 


CONTENTS 


Explanation  of  the  Tables 


TABLES   PRINCIPALLY   TO   FIVE   PLACES 


Table 

I. 

Table 

la 

Table 

11. 

Table 

Ilia. 

Table 

III. 

Table 

IV. 

Table 

V. 

Table 

Va. 

Table 

VI. 

Table 

VII. 

Table  VIII. 

Table 

IX. 

Table 

X. 

Table 

XI. 

Table 

Xlla. 

Table 

XII&. 

Table 

XIIc. 

Table 

Xlld. 

Table 

Xlle. 

Table 

XII/. 

Table  XIII. 

Common  Logarithms  of  Numbers    . 
Condensed  Logarithms  and   Antilogarithms 
Actual  Values  of  the  Trigonometric  Func- 
tions     

Values  of  S  and  T  for  Interpolation    . 
Common   Logarithms    of   the  Trigonometric 
Functions     ...... 

Reduction  of  Degrees  to  Radians 
Trigonometric  Functions  in  Radian  Measure 
Reduction  of  Radians  to  Degrees 
Powers  —  Roots  —  Reciprocals 
Napierian  or  Natural  Logarithms 
Multiples  of  M  and  of  1/M  .  . 
Values    and     Logarithms     of     Hyperbolic 

Functions 

Values  and  Logarithms  of  Haversines 
Factor  Table  —  Logarithms  of  Primes 
Compound  Interest  ..... 
Compound  Discount  .... 

Amount  of  an  Annuity    .... 
Present  Value  of  an  Annuity 
Logarithms  for  Interest  Computations 
American  Experience  Mortality  Table 
Important  Constants         .... 


BRIEF   TABLES  — PRINCIPALLY   TO   FOUR   PLACES 


Table  XlVa.  Common  Logarithms 134-135 

Table  XIV6.  Antilogarithms 136-137 

Table  XIVc.    Values  and   Logarithms   of   Trigonometric 

Functions 138-142 


EXPLANATION   OF  THE   TABLES* 

TABLE  I.     FIVE-PLACE   COMMON   LOGARITHMS  OF 
NUMBERS   FROM  1   TO   10  000 

1 .  Powers  of  10.     Consider  the  following  table  of  values  of  powers  of  10: 


Column  A 

Column  £ 

Column  A 

Column  B 

101 

= 

10 

100 

= 

1. 

102 

= 

100 

10-1 

= 

.1 

103 

= 

1000 

10-2 

= 

.01 

104 

= 

10000 

10-3 

= 

.001 

105 

=: 

100000 

10-4 

= 

.0001 

106 

= 

1000000 

10-5 

.00001 

107 

= 

10000000 

10-6 

_ 

.000001 

108 

= 

100000000 

10-7 

= 

.0000001 

109 

= 

1000000000 

10-8 

= 

.00000001 

1010 

= 

10000000000 

10-9 

= 

.000000001 

This  table  may  be  used  for  multiplying  or  dividing  powers  of  10,  by 
means  of  the  rules  10«  •  10»  =  10«+^  10«  -f-  10»  =  10«-^  Thus,  to  multiply 
1000  by  100,000,  add  the  exponent  of  10  in  column  J.  opposite  1000  to  the 
exponent  of  10  opposite  100,000 :  3+5=8;  and  look  for  the  number  in 
column  B  opposite  108,  ^-^g.  100,000,000.  Similarly  1,000,000  x  .0001  =  100, 
since  6+  (—4)  =2. 

To  divide  1,000,000  by  100,  from  the  exponent  of  10  opposite  1,000,000 
subtract  the  exponent  of  10  opposite  100 ;  6  —  2=4;  and  look  for  the 
number  opposite  10*,  i.e.  10,000.  Similarly  .001  h- 1,000,000  =  .000000001, 
since  —  3  —  6  =  —  9.  To  find  the  4th  power  of  100,  multiply  the  exponent 
of  10  opposite  100  by  4  :  4x2  =  8,  and  look  for  the  number  opposite  108, 
i.e.  100,000,000.  Likewise  (.001)3  =  .000000001,  since  3  x  (-  3J  =-  9. 
To  find  the  cube  root  of  1,000,000,000,  divide  the  exponent  of  10  opposite 
1,000,000,000  by  3,  9-4-3  =  3,  and  look  for  the  number  opposite  103. 


*  This  Explanation,  written  to  accompany  the  five-place  tables,  may  be  used  also  for  the 
four-place  tables  by  omitting  the  last  figure  in  each  example  in  a  manner  obvious  to  the 
teacher. 

vii 


VUl 


EXPLANATION  OF  THE  TABLES 


[§2 


2.  Common  Logarithms.  The  exponent  of  10  in  any  row  of  column  A 
is  called  the  common  logarithm  *  of  the  number  opposite  in  column  B ; 
thus  log  10  =  1,  log  100  =  2,  log  1000  =  3,  etc.;  log  1  =  0,  log  .1  =-  1  ; 
log  .01  =—2,  log  .001  =—3,  etc.  In  general,  if  10^  =  n,  Z  is  called  the 
common  logarithm  of  w,  and  is  denoted  by  log  n. 

3.  Fundamental  Principles.  Logarithms  are  useful  in  reducing  the 
labor  of  performing  a  series  of  operations  of  multiplication,  division, 
raising  to  powers,  extracting  roots,  as  above  ;  they  have  no  necessary 
connection  with  trigonometry,  since  all  the  operations  could  be  performed 
without  them  ;  but  they  are  a  great  labor-saving  device  in  arithmetical 
computations.     They  do  not  apply  to  addition  and  subtraction. 

The  principles  of  their  application  are  stated  as  follows  : 

I.  The  logarithm  of  a  product  is  equal  to  the  sum  of  the  logarithms  of 
the  factors :  log  ab  =  log  a  +  log  b.  This  follows  from  the  fact  that  if 
10^  =  a  and  10^  =  6,  10^+^  =  a  •  &.   In  brief :  to  multiply,  add  logarithms. 

II.  The  logarithm  of  a  fraction  is  equal  to  the  difference  obtained  by 
subtracting  the  logarithm  of  the  denominator  from  the  logarithm  of  the 
numerator :  log  {a/b)  =  log  a  —  log  b.  For,  if  10^  =  a  and  10^  —  b,  then 
lOi-L  _  ^  _^  ^^    jji  i^rief  :  to  divide,  subtract  logarithms. 

III.  The  logarithm  of  a  power  is  equal  to  the  logarithm  of  the  base 
multiplied  by  the  exponent  of  the  power :  log  a^  =  b  log  a.  This  follows 
from  the  fact  that  if  10^  =  a,  then  W^  -  ap. 

IV.  The  logarithm  of  a  root  of  a  number  is  found  by  dividing  the  loga- 
rithm of  the  number  by  the  index  of  the  root:  log  Va  =  (log  a)/&.  This 
follows  from  the  fact  that  if  W  =  a,  then  lOV^  =  «!/&  =  \/a. 

Corollary  of  II.  The  logarithm  of  the  reciprocal  of  a  number  is  the 
negative  of  the  logarithm  of  the  number :  log  (1/a)  =  —  log  a,  since 
log  1  =  0. 

4.  Characteristic  and  Mantissa.  It  is  shov^n  in  algebra  that  every 
real  positive  number  has  a  real  common  logarithm,  and  that  if  a  and  b 
are  any  two  real  positive  numbers  such  that  a  <  6,  then  log  a  <  log  b. 
Neither  zero  nor  any  negative  number  has  a  real  logarithm. 

An  inspection  of  the  following  table,  which  is  a  restatement  of  a  part 


a 

1 

10 

100 

1000 

10000 

100000 

1000000 

10000000 

log  a 

0 

1 

2 

3 

4 

5 

6 

7 

*  Common  logarithms  are  exponents  of  the  base  10 ;  other  systems  of  logarithms  have 
bases  different  from  10 ;  Napierian  logarithms  (see  Table  VII,  p.  112)  have  a  base  denoted  by 
e,  an  irrational  number  whose  value  is  approximately  2.71828.  When  it  is  necessary  to  call 
attention  to  the  base,  the  expression  log^o  n  will  mean  common  logarithm  of  n  ;  loge  n  will 
mean  the  Napierian  logarithm,  etc. ;  but  in  this  book  log  n  denotes  logjo^  unless  otherwise 
explicitly  stated. 


^4] 


COMMON  LOGARITHMS 


IX 


of  the  table  of  §  1,  p.  v,  shows  that 
the  logarithm  of  every  number  between  1  and  10  is  a  proper  fraction, 
the  logarithm  of  every  number  between  10  and  100  is  1  -f  a  fraction, 
the  logarithm  of  every  number  between  100  and  1000  is  2  +  a  fraction  ; 
and  so  on.     It  is  evident  that  the  logarithm  of  every  number  (not  an 
exact  power  of  10)   consists  of  a  whole  number  +  a  fraction  (usually 
written  as  a  decimal).     The  whole  number  is  called  the  characteristic; 
the  decimal  is  called  the  mantissa.     The  characteristic  of  the  logarithm 
of  any  number  greater  than  1  may  be  determined  as  follows  : 

Rule  I.     The  characteristic  of  any  number  greater  than  1  is  one  less 
than  the  number  of  digits  before  the  decimal  point. 

The  following  table,  which  is  taken  from  §  1,  p.  v,  shows  that 


a 

.0000001 

.000001 

.00001 

.0001 

.001 

.01 

.1 

1 

log  a 

-7 

-6 

-5 

-4 

~3 

—  2 

-  1 

0 

the  logarithm  of  every  number  between  .1  and  1  is  —  1  +  a  fraction, 
the  logarithm  of  every  number  between  .01  and  .1  is  —  2  +  a  fraction, 
the  logarithm  of  every  number  between  .001  and  .01  is  —  3  +  a  fraction  ; 
and  so  on. 

Thus  the  characteristic  of  every  number  between  0  and  1  is  a  negative 
whole  number ;  there  is  a  great  practical  advantage,  however,  in  comput- 
ing, to  write  these  characteristics  as  follows :  —  1  =  9  —  10,  —  2  =  8  —  10, 
—  3  =  7  —  10,  etc.  E.g.  the  logarithm  of  .562  is  -  1  +  .74974,  but  this 
should  be  written  9.74974  —  10  ;  and  similarly  for  all  numbers  less  than  1. 
Rule  II,  The  characteristic  of  a  number  less  than  1  is  found  by  sub- 
tracting from  9  the  number  of  ciphers  between  the  decimal  point  and  the 
first  significant  digits  and  writing  —  10  after  the  result. 

Thus,  the  characteristic  of  log  845  is  2  by  Rule  I ;  the  characteristic  of 
log 84.5  is  1  by  (I)  ;  of  log8.45  is  0  by  (I)  ;  of  log. 845  is  9  -  10  by  (II)  ; 
of  log. 0845  is  8  -  10  by  (II). 
An  important  consequence  of  what  precedes  is  the  following  : 
To  move  the  decimal  point  in  a  given  number  one  place  to  the  right  is 
equivalent  to  adding  one  unit  to  its  logarithm,  because  this  is  equivalent 
to  multiplying  the  given  number  by  10.     Likewise,  to  move  the  decimal 
point  one  place  to  the  left  is  equivalent  to  subtracting  one  unit  from  the 
logarithm.     Hence,  moving  the  decimal  point  any  number  of  places  to 
the  right  or  left  does  not  change  the  mantissa  but  only  the  characteristic* 
Thus,  5345,  5.345,  534.6,  .05345,  534500  all  have  the  same  mantissa. 


*  Another  rule  for  finding  the  characteristic,  based  on  this  property,  is  often  useful : 
if  the  decimal  point  were  just  after  the  first  significant  figure,  the  characteristic  would  be 
zero  ;  start  at  this  point  and  count  the  digits  passed  over  to  the  left  or  right  to  the  actual 
decimal  point ;  the  number  obtained  is  the  characteristic,  except  for  sign  ;  the  sign  is  nega- 
tive if  the  movement  was  to  the  left,  positive  if  the  movement  was  to  the  right. 


X  EXPLANATION  OF  THE  TABLES  [§  5 

5.  Use  of  the  Table.  To  use  logarithms  in  computation  we  need  a 
table  arranged  so  as  to  enable  us  to  find,  with  as  little  effort  and  time  as 
possible,  the  logarithms  of  given  numbers  and,  vice  versa,  to  find  numbers 
when  their  logarithms  are  known.  Since  the  characteristics  may  be 
found  by  means  of  llules  I  and  II,  p.  ix,  only  mantissas  are  given.  This 
is  done  in  Table  I.  Most  of  the  numbers  in  this  table  are  irrational,  and 
must  be  represented  in  the  decimal  system  by  approximations.  A  five- 
place  table  is  one  which  gives  the  values  correct  to  five  places  of  decimals. 

Problem  1.  To  Ji7id  the  logarithm  of  a  given  number.  First,  deter- 
mine the  characteristic,  then  look  in  the  table  for  the  mantissa. 

To  find  the  mantissa  in  the  table  when  the  given  number  (neglecting 
the  decimal  point)  consists  of  four,  or  less,  digits  (exclusive  of  ciphers  at 
the  beginning  or  end),  look  in  the  column  marked  iVfor  the  first  three 
digits  and  select  the  column  headed  by  the  fourth  digit :  the  mantissa 
will  be  found  at  the  intersection  of  this  row  and  this  column.  Thus  to 
find  the  logarithm  of  72050,  observe  first  (Eule  I)  that  the  characteristic 
is  4.  To  find  the  mantissa,  fix  attention  on  the  digits  7205  ;  find  720  in 
column  iV,  and  opposite  it  in  column  5  is  the  desired  mantissa,  .85763 ; 
hence  log  72050  =  4.85763.  The  mantissa  of  .007826  is  found  opposite 
782  in  column  6  and  is  .89354 ;  hence  log  .007826  =  7.89354—  10. 

6.  Interpolation.  If  there  are  more  than  four  significant  figures  in  the 
given  number,  its  mantissa  is  not  printed  in  the  table ;  but  it  can  be 
found  approximately  by  assuming  that  the  mantissa  varies  as  the  number 
varies  in  the  small  interval  not  tabulated ;  while  this  assumption  is  not 
strictly  correct,  it  is  sufficiently  accurate  for  use  with  this  table. 

Thus,  to  find  the  logarithm  of  72054  we  observe  that  log  72050  =  4.85763 
and  that  log  72060  =  4.85769.  Hence  a  change  of  10  in  the  number  causes 
a  change  of  .00006  in  the  mantissa ;  we  assume  therefore  that  a  change  of 
4  in  the  number  will  cause,  approximately,  a  change  of  .4  x  .00006 
=  .00002  (dropping  the  sixth  place)  in  the  mantissa ;  and  we  write 
log  72054  =  4.85763  +  .00002  =  4.85765. 

The  difference  between  two  successive  values  printed  in  the  table  is 
called  a  tabular  difference  (.00006,  above).  The  proportional  part  of 
this  difference  to  be  added  to  one  of  the  tabular  values  is  called  the  cor- 
rection (.000002,  above),  and  is  found  by  multiplying  the  tabular  difference 
by  the  appropriate  fraction  (.4,  above).  These  proportional  parts  are 
usually  written  without  the  zeros,  and  are  printed  at  the  right-hand  side 
of  each  page,  to  be  used  when  mental  multiplications  seem  uncertain. 

Example  1.  Find  the  logarithm  of  .0012647.  Opposite  126  in  column  4  find  .10175; 
the  tabular  difference  is  34  (zeros  dropped)  ;  .7  x  34  is  given  in  the  margin  as  24 ;  this  cor- 
rection added  gives  .10199  as  the  mantissa  of  .0012647 ;  hence  log  .0012647  =  7.10199  -  10. 

Example  2.  Find  the  logarithm  of  1.85643.  Opposite  185  in  column  6  find  .26858 ; 
tabular  difference  23  ;  .43  x  23  is  given  in  the  margin  as  10 ;  this  correction  added  gives 
.26868  as  the  mantissa  of  1.86643  ;  hence  log  1.85643=  0.26868. 


§8]  COMMON    LOGARITHMS  xi 

7.  Reverse  Reading  of  the  Table.  Problem  2.  To  find  the  number 
when  its  logarithm  is  known.*  First,  fixing  attention  on  the  mantissa 
only,  find  from  the  table  the  number  having  this  mantissa,  then  place  the 
decimal  point  by  means  of  the  two  following  rules  :  t 

Rule  III.  If  the  characteristic  of  the  logarithm  is  positive  (in  which 
case  the  mantissa  is  not  followed  by  —  10),  begin  at  the  left,  count  digits 
one  more  than  the  characteristic,  and  place  the  decimal  point  to  the  right 
of  the  last  digit  counted. 

Rule  IV.  If  the  characteristic  is  negative  (in  which  case  the  mantissa 
will  be  preceded  by  a  number  n  and  followed  by  —  10),  prefix  9—  n 
ciphers,  and  place  the  decimal  point  to  the  left  of  these  ciphers. 

Example  1.     Given  log  x  =  1.22737,  to  find  x. 

Since  the  mantissa  is  22737,  we  look  for  22  in  the  first  column  and  to  the  right  and  below 
for  737,  which  we  find  in  column  8  opposite  168.  The  number  is  therefore  1688.  Since  the 
characteristic  is  + 1,  we  begin  at  the  left,  count  2  places,  and  place  the  point ;   hence 

X  =  16.88. 

Example  2.    Given  log  x  =  2.24912,  to  find  x. 

This  mantissa  is  not  found  in  the  table  ;  in  such  cases  we  interpolate  as  follows  :  select 
the  mantissa  in  the  table  next  less  than  the  given  mantissa,  and  write  down  the  corre- 
sponding number ;  here,  1774 ;  the  tabular  difference  is  25  ;  the  actual  difiference  (found  by- 
subtracting  the  mantissa  of  1774  from  the  given  mantissa)  is  17  ;  hence  the  proportionality- 
factor  is  17/25  =  .68  or  .7  (to  the  nearest  tenth).  Since  moving  the  decimal  point  does  not 
affect  the  mantissa,  it  follows  that  the  digits  in  the  required  number  are  17747  (to  five  places). 
The  characteristic  2  directs  to  count  3  places  from  the  left ;  hence  x  =  177.47. 

Rule.  In  general,  when  the  given  mantissa  is  not  found  in  the  table, 
write  down  four  digits  of  the  number  corresponding  to  the  mantissa  in  the 
table  next  less  than  the  given  mantissa,  determine  a  fifth  figure  by  dividing 
the  actual  difference  by  the  tabular  difference,  and  locate  the  decimal  point 
by  means  of  the  characteristic. 

8.  Illustrations  of  the  Use  of  Logarithms  in  Computation. 

Example  1.    To  find  832.43  X  302.43  X  16.725  X  .000178. 
log  832.43  =  2.92034 
log  302.43  =  2.48062 
log  16.725  =  1.22337 
log  .000178  =  6.25042  -  10  (add) 

log  X  =  2.87475  whence  x  —  749.47. 

Example  2.    To  find  461.29  ^  21.4. 

log  461.29  =  2.66397 
log  21 .4  =  1.33041  (subtract) 

log  X  =  1.33356  whence  x  =  21.556. 


*  The  number  whose  logarithm  is  k  is  often  called  the  antilogarithm  oik. 

t  Another  convenient  form  of  these  rules  is  as  follows :  if  the  characteristic  were  zero, 
the  decimal  point  would  fall  just  after  the  first  significant  figure ;  move  the  decimal  point 
one  place  to  the  right  for  each  positive  unit  in  the  characteristic,  one  place  to  the  left  for 
each  negative  unit  in  the  characteristic. 


xii  EXPLANATION  OF  THE  TABLES  [§8 

Illustration  of  Cologarithms 

E^mpUZ.    Tofindl5^25xm76XJT45. 

1415.3 
We  might  add  the  logarithms  of  the  factors  in  the  numerator  and  from  this  sum  subtract 
the  logarithm  of  the  denominator ;  but  we  can  shorten  the  operation  by  adding  the  nega- 
tive of  the  logarithm  of  the  denominator  instead  of  subtracting  the  logarithm  itself.  The 
negative  of  the  logarithm  of  a  number  (when  written  in  convenient  form  for  computation) 
is  called  the  cologarithm  of  the  number.  We  may  find  the  negative  of  any  number  by 
subtracting  it  from  zero,  and  it  is  convenient  in  logarithmic  computation  to  write  zero  in  the 
form  10.00000  -  10.  Thus  the  negative  of  2.17  is  7.83  -  10 ;  the  negative  of  1.1432  -  10  is 
8.8568.  Remembering  that  the  cologarithm  of  a  number  is  its  negative  we  have  the  follow- 
ing rule : 

To  find  the  cologaHthm  of  a  nuniber  hegin  at  the  left  of  its  logarithm  {including 
the  characteristic)  and  subtract  each  digit  from  9,  except  the  last,*  which  subtract 
from  10  ;  if  the  logarithm,  has  not  —  10  after  the  mantissa^  write  —  10  after  the  result; 
if  the  logarithm  has  —  10  after  the  mantissa,  do  not  write  —  10  after  the  result. 

By  this  rule  the  cologarithm  of  a  number  can  be  read  directly  out  of  the  table  without 
taking  the  trouble  to  write  down  the  logarithm.    Attention  must  be  given  not  to  forget  the 
characteristic.    The  use  of  the  cologarithm  is  governed  by  the  principle  : 
Adding  the  cologarithm  is  equivalent  to  subtracting  the  logarithm, 
Eeturning  to  the  computation  of  the  given  problem  we  should  write : 
log  48. 25  =1.68350 
log  132.76=  2.12307 
log  .1745=  9.24180 -10 
colog  1415.3  =  6.84915  -  10    (add) 

log  x=  9.89752  -  10    whence  x=  .7898 
Esaample  4.    Find  the  5th  power  of  7.26842 

log  7.26842=  0.86144 

5    (multiply) 

log  X  =  4.30720    whence  x  =  20286. 

Example  5.  Find  the  4th  root  of  .007564 

log  .007564  =7.87875 -10. 
(It  is  convenient  to  have,  after  the  division  by  4s  —  10  after  the  mantissa ;  hence  before  the 
division  we  add  30.00000  -  30.) 

log  .007564=  37.87875  -  40    (divide  by  4), 

log  X  =   9.46969  -  10    whence  x  =  .2949 


Example^.    Find  the  value  of  ! /(34.55)(-  856.7)(-  43]!) 
\         (98.75)(- 186.3) 


We  have  no  logarithms  of  negative  numbers,  but  an  inspection  of  this  problem  shows 
that  the  result  will  be  negative  and  numerically  the  same  as  though  all  the  factors  wer« 
positive ;  hence  we  proceed  as  follows : 

log  34.55  =1.53845 
log  856.7  =  2.93288 
log  43.5  =1.63849 
colog  98.75=  8.00546  -  10 
colog  186.3  =  7.72979  -  10        (add) 

1.84502  (divide  by  3) 

log(-  a;)  =  0.61501  whence  tc  =  -  4.121 


*  If  the  logarithm  ends  in  one  or  more  ciphers,  the  last  significant  digit  is  to  be  under 
Btood  here. 


§9] 


THE  SLIDE   RULE 


Xlll 


9.   The  Slide  Rule.    A  slide  rule  consists  of  two  pieces  of  the  shape 
of  a  ruler,  one  of  which  slides  in  grooves  in  the  other ;  each  is  marked 


3'^ 


6      7     8    9    1 


Mnirni 


IMIMltr 


lllllillllllllll 


Fig.  1 

(Fig.  1)  in  divisions  (scale  A  and  scale  B)  whose  distances  from  one  end 
are  proportional  to  the  logarithms  of  the  numbers  marked  on  them. 
It  follows  that  the  sum  of  two  logarithms  can  be  obtained  by  simply 


1                            2                S'^        4567891 

A  1                                 1           '         '  '           '           !         1        1       1      1     i 

2 

^hiiliiili  ihliiiiililililil:'^''           '        '    '  '      '  :  ''ih 

thtr 

Ttltltltlltlll]] 

'  .  :  -nll|llll|llll!llll|llll|lll'!!lll|li| 

A 

L                 1            2          '       3^         4 

5       6      7    8    91 

; 

2 

K  ,  ,  M 

/ 

m 

IJTT It  Ti- 

mill 

Wm\l 

ti: 

,  ,  ,1,  , 

Vi'.lW 

pJIIII  Nil  Nil  lllllllllll 

1   1 

il.L 

ll                  II 

II   1 

III 

4|ii 

'\'i 

U  lU 

D^ 1 [ \ 1         1        '       '       ■       ■     ^     :      ■      :     1     j     ■    .     :    .^i    ■    ■    . 

V 

^'i""'"' 

Fig.  2 

sliding  one  rule  along  the  other ;  thus  if  (see  Fig.  2)  the  point  marked  1 
on  scale  B  is  set  opposite  the  point  marked  2. 5  on  scale  A^  the  point  on 
scale  B  marked  2  will  be  opposite  the  point  on  scale  A  marked  6,  since 
log  2.5  +  log  2  =  log  5.  Likewise,  opposite  3  (scale  B)  read  7.6  (scale  A)  \ 
opposite  2.5  (J5)  read  6.25  (^),  i.e.  2.5  x  2.6  =  6.25. 

Other  multiplications  can  be  performed  in  an  analogous  manner.  Divi- 
sions can  be  performed  by  reversing  the  operation.  Thus,  if  4.5  {B)  be 
set  on  11.25  (^),  then  1  {B)  will  be  opposite  2.5  (J.),  as  in  Fig.  2. 

Scales  C  and  J)  are  made  just  twice  as  large  as  scales  A  and  B.  It  fol- 
lows that  the  numbers  marked  on  0  and  B  are  the  square  roots  of  the 
numbers  marked  opposite  them  on  scales  ^  and  B. 

For  a  description  of  more  elaborate  slide  rules,  and  full  directions  for 
use,  see  the  catalogues  of  instrument  makers. 

A  slide  rule  for  practice  may  be  made  from  the  cut  printed  on  one  of 
the  fly-leaves  in  the  back  of  this  book. 


xiv  EXPLANATION  OF  THE  TABLES  [§  10 

la.   CONDENSED    LOGARITHMS    AND    ANTILOGARITHMS 

10.  Method  of  Computing  Logarithms.  This  table  is  a  rearrangement 
of  the  condensed  table  given  by  Hoiiel.*  From  it,  the  logarithm  of  any 
number  whatever  may  be  obtained  to  within  5  in  the  fifteenth  place  ;  or 
to  any  desired  degree  of  accuracy  less  than  this. 

To  illustrate  the  process,  we  shall  compute  log  w  to  nine  places.  Tak- 
ing TT  =  3.1415926535  8979,  we  divide  it  by  3,  the  first  significant  digit, 
obtaining  7r/3  =  1.04719  755  •••.  We  then  divide  this  quotient  by  1.04, 
etc.,  obtaining  finally 

TT  =  3(1.04)  (1.006)  (1.0009)  (1.00001  5217225). 
We  can  obtain  the  logarithm  of  each  of  the  first  four  factors  from  this 
table.    The  logarithm  of  the  last  factor  can  be  obtained  by  multiplying 
its  decimal  part  hjM=  .4342944819  ;  for  the  error  made  in  writing 

log(l  +x)  =  Mx 
is  less  than  Mx'^/2.     We  find  Mx  either  by  using  the  fact  that  the  last 
column  in  this  table  gives  multiples  of  If,  or  (preferably)  by  Table  VIII, 
page  115.     Adding  the  five  logarithms  just  mentioned,  we  find 

log7r=  .4971498727  4, 
which  is  surely  correct  to  within  1  in  the  tenth  place.     The  correct  value 
is  .4971498726  9  .... 

The  process  may  be  applied  to  any  other  number  in  an  analogous  man- 
ner. Such  high-place  logarithms  are  occasionally  needed  in  statistical 
work  and  in  the  preparation  of  tables. 

11.  Method  of  Computing  Antilogarithms.  The  condensed  table  of 
antilogarithms  gives  eleven  significant  figures  (ten  decimal  places).  From 
it,  the  antilogarithm  of  any  number  can  be  computed  to  within  6  in  the 
tenth  significant  digit. 

Thiis,  to  compute  the  antilogarithm  of  .4342944819  to  8  significant 
figures,  we  may  write 

10-4342944819  —  (10-4)  (lO-^^)  (10-004)  (10•0002^  H  0.00009)  (10.0000044819) . 

The  first  five  factors  may  be  obtained  directly  from  the  table.  The  last 
factor  may  be  calculated  from  the  formula  10*  =  1-1-  {\/M)x.  The  error 
in  this  formula  is  less  than  3  in  the  (2  A:)th  decimal  place  if  x  is  less  than 
(.1)*,  where  A:>1. 

However,  a  much  more  rapid  process  depends  on  the  use  of  Tables  I  and 
XI  with  this  table.  Thus,  by  Table  I,  10-43429  _  2. 718,  nearly.  By  Table 
XI,  log  2.718  =  .43424  94524  ....  Hence  10.4342944819  =^(2.718) (10-0000450295) 
=  (2.718)  (10-00004)  (10-0000050296).  Obtaining  the  second  factor  from  this 
table,  and  the  last  factor  from  the  formula  10'  =  1  4-  (l/if)x,  by  Table 
VHI,  we  find  10-4342944819=^2.718281826;  while  the  correct  value  is 
2.718281828  •••.     This  process  requires  only  two  long  multiplications. 

*  HotJEL,  Becueil  de  Formules  ei  de  Tables  numiriques. 


§  12]  TRIGONOMETRIC  FUNCTIONS  XV 

II.     FIVE-PLACE   TABLE   OF    THE   ACTUAL   VALUES   OF 
THE  TRIGONOMETRIC   FUNCTIONS  OF  ANGLES 

12.  Direct  Readings.  This  table  gives  the  sines,  cosines,  tangents, 
and  cotangents  of  the  angles  from  0°  to  45°  ;  and  by  a  simple  device, 
indicated  by  tlie  printing,  the  values  of  these  functions  for  angles  from 
45°  to  90°  may  be  read  directly  from  the  same  table.  For  angles  less  than 
45°  read  down  the  page,  the  degrees  being  found  at  the  top  and  the  min- 
utes on  the  left ;  for  angles  greater  than  45°  read  up  the  page,  the  degrees 
being  found  at  the  bottom  and  the  minutes  on  the  right. 

To  find  a  function  of  an  angle  (such  as  15°27'.6,  for  example)  v^hich 
does  not  reduce  to  an  integral  number  of  minutes,  we  employ  the  process 
of  interpolation.  To  illustrate,  let  us  find  tan  15°  27'. 6.  In  the  table 
we  find  tan  15°  27' =  .27638  and  tan  15°  28' =  .27670  ;  we  know  that 
tan  15°  27 '.6  lies  between  these  two  numbers.  The  process  of  interpola- 
tion depends  on  the  assumption  that  between  15°  27'  and  15°  28'  the  tan- 
gent of  the  angle  varies  directly  as  the  angle  ;  while  this  assumption  is  not 
strictly  true,  it  gives  an  approximation  sufficiently  accurate  for  a  five-place 
table.  Thus  we  should  assume  that  tan  15°  27'. 5  is  halfway  between 
.27638  and  .27670.  We  may  state  the  problem  as  follows  :  An  increase 
of  1'  in  the  angle  increases  the  tangent  .00032  ;  assuming  that  the  tangent 
varies  as  the  angle,  an  increase  of  0'.6  in  the  angle  will  increase  the  tan- 
gent by  .6  X  .00032  =  .00019  (retaining  only  five  places);  hence 
tan  15°  27'.6  =  .27638  +  .00019  =  .27657. 

The  difference  between  two  successive  values  in  the  table  is  called,  as 
in  Table  I,  the  tabular  difference  (.00032  above).  The  proportional  part 
of  the  tabular  difference  which  is  used  is  called  the  correction  (.00019 
above),  and  is  found  by  multiplying  the  tabular  difference  by  the  appro- 
priate fraction  of  the  smallest  unit  given  in  the  table. 

Example  1 .    Find  sin  63°  52 '  .8. 

We  find  sin  63*'i52  '  =  .  89777 ; 

tabular  difference  =  .00013  (subtracted  mentally  from  the  table), 
correction  =  .8  x  .00013=  .00010  (to  be  added). 
Hence  sin  63"  62'.8  =  .89787. 

ExampU  2.    Find  cos  65°  24'.8. 

cos  65°  24'  =  .41628 ; 
tabular  difference  =  26 ;  .8  x  26  =  21 
(to  be  subtracted  because  the  cosine  decreases  as  the  angle  increases) . 
Hence  cos  65°  24'. 8  =  .41607. 

Rule.  To  find  a  trigonometric  function  of  an  angle  by  interpolation  : 
select  the  angle  in  the  table  which  is  next  smaller  than  the  given  angle,  and 
read  its  sine  (cosine  or  tangent  or  cotangent  as  the  case  may  be)  and  the 
tabular  difference.  Compute  the  correction  as  the  proper  proportional 
part  of  the  tabular  difference.  In  case  of  sines  or  tangents  add  the  correc- 
tion ;  in  case  of  cosines  or  cotangents,  subtract  it. 


xvi  EXPLANATION  OF  THE   TABLES  [§  13 

13.  Reverse  Readings.  Interpolation  is  also  used  in  finding  the  angle 
when  one  of  its  functions  is  given. 

Example  1.     Given  sin  a?=  .32845,  to  find  x. 

Looking  in  the  table  we  find  the  sine  which  is  next  less  than  the  given  sine  to  be  .32832, 
and  this  belongs  to  19*'10'.  Subtract  the  value  of  the  sine  selected  from  the  given  sine  to 
obtain  the  actual  difi'erence=  ,00013  ;  note  that  the  tabular  difference  =  ,00027.  The  actual 
difference  divided  by  the  tabular  difference  gives  the  correction  =>  13/27  =  .5  as  the  decimal 
of  a  minute  (to  be  added).     Hence  x—  19**  10'. 5. 

Example  2.    Given  cos  x=  .28432,  to  find  x. 

The  cosine  in  the  table  next  less  than  this  is  .28429  and  belongs  to  73**  29' ;  the  tabular 
difference  is  28;  the  actual  difference  is  3;  correction  =  3/28=  .1  (to  be  subtracted). 
Hence  85  =  73"  28 '.9. 

KuLE.  To  find  an  angle  when  one  of  its  trigonometric  functions  is  given  : 
select  from  the  table  the  same  named  function  which  is  next  less  than  the 
given  function,  noting  the  corresponding  angle  and  the  tabular  difference ; 
compute  the  actual  difference  (between  the  selected  value  of  the  function 
and  the  given  value)  and  divide  it  by  the  tabular  difference  ;  this  gives  the 
correction  which  is  to  be  added  if  the  given  function  is  sine  or  tangent, 
and  to  be  subtracted  if  the  given  function  is  cosine  or  cotangent. 

in.     FIYE-PLACE   COMMON   LOGARITHMS   OF   THE 
TRIGO]N^OMETRIC   FUNCTIONS 

14.  Use  of  the  Table.  If  it  is  required  to  find  the  numerical  value  of 
X  =  27.85  X  sin  51°  27',  we  may  apply  logarithms  as  follows  : 

log27.85  =  1.44483. 
log  sin  51°  27'  =  9.89324  -  10  (add) . 

logx  =  1.33807  x  =  21.78 

The  only  new  idea  here  is  the  method  of  finding  log  sin  51°  27',  which 
means  the  logarithm  of  the  sine  of  51°  27'.  The  most  obvious  way  is  to  find 
in  Table  I,  sin 51° 27'  =  .78206,  and  then  to  find  in  Table  II,  log. 78206 
=  9.89324  —  10,  but  this  involves  consulting  two  tables.  To  avoid  the 
necessity  of  doing  this.  Table  HI  gives  the  logarithms  of  the  sines, 
cosines,  tangents,  and  cotangents.  The  arrangement  and  the  principles 
of  interpolation  are  similar  to  those  given  on  p.  viii  for  Table  I.  The  sines 
and  cosines  of  all  acute  angles,  the  tangents  of  all  acute  angles  less  than  45° 
and  the  cotangents  of  all  acute  angles  greater  than  45°  are  proper  fractions, 
and  their  logarithms  end  with  —  10,  which  is  not  printed  in  the  table,  but 
which  should  be  written  down  whenever  such  a  logarithm  is  used. 

Example  1.    Find  log  sin  68°  25'. 4. 

On  the  page  having  68**  at  the  bottom,  and  in  the  row  having  25'  on  the  right  find  log 
sin  68°  25'  =  9.96843  -  10 ;  the  tabular  difference  is  5 ;  .4  x  5  is  given  in  the  margin  as  2 ; 
this  is  the  correction  to  be  added,  giving  log  sin  68°  25'. 4=  9.96845  -  10. 

(In  case  of  sine  and  tangent  add  the  correction.  In  case  of  cosine  and  cotangent,  sub- 
tract the  correction.) 


§  15]  RADIAN  MEASURE  xvii 

Example  2.    Given  log  cos  a;  =  9.72581  —  10,  to  find  x. 

The  logarithmic  cosine  next  less  than  the  given  one  is  9.72562—10  and  belongs  to  57"  53' ; 
the  actual  difference  is  19  ;  the  tabular  difference  is  20 ;  hence  the  correction  is  19/20=  1.0 
(to  the  nearest  tenth)  ;  (subtract)  ;  hence  x=  57"  52'. 0. 

In  finding  log  ctn  a  for  any  angle  a,  note  that  log  ctn  ct  =  —  log  tan  a, 
since  ctn  a  =  1 /tan  a.  Hence  the  tabular  differences  for  log  ctn  are  pre- 
cisely the  same  as  those  for  log  tan  throughout  the  table,  but  taken  in 
reversed  order.  Likewise,  log  sec  a  =—  log  cos  a,  log  esc  cc  =  —  log  sin  a  ; 
hence  log  sec  a  and  log  esc  a  are  omitted. 

For  angles  near  0°  or  near  90°,  the  interpolations  are  not  very  accurate 
if  the  differences  are  large.  For  the  calculation  of  sine  or  tangent  near 
0°,  Table  Ilia,  page  45,  gives  the  values  of 

S  =  log  sin  A  —  log  A'        and         T  =  log  tan  A  —  log  A', 
where  A  is  the  given  angle  and  A'  is  the  number  of  minutes  in  A,  for 
values  of  J.  between  0°  and  3°.    Then 

log  sin  A  =  \ogA'  -^  S        and        log  tan  A  =  log  A'  +  T, 
for  small  angles.     Moreover,  since  we  have  cos  J.  =:  sin  (90°  —  J.)    and 
ctnJ.  =  tan(90°- J.), 

log  cos  J.  =  log  (90°  -  ^)'4-  ^  and  log  ctn  J.  =  log  (90°  -  A)' +  T, 
when  A  is  near  90°. 

Another  method  practically  equivalent  to  the  preceding  is  to  use  the 
approximate  relations 

log  sin  A  —  log  sin  B  =  log  A'  —  log  B' 
and 

log  tan  A  —  log  tan  B  =  log  A^  —  log  B', 

where  A  is  the  given  angle  and  B  is  the  nearest  angle  to  A  that  is  given 
in  the  table.  If  J.  <  3°  and  \A  —  5 1  <  1',  these  formulas  give  log  sin  A 
and  log  tan  A  to  five  decimal  places. 

IV-y.     RADIAN  MEASURE 

15.  Computations  in  Radian  Measure.  The  reduction  of  degrees  to 
radians  is  facilitated  by  Table  TV —  Conversion  of  Degrees  to  Radians. 
Since  tt  radians  =  180°,  this  table  may  be  regarded  as  a  table  of  multiples 
of  7r/180. 

The  values  of  sinx,  coscc,  tana;,  are  stated  for  every  angle  x  from  0.00 
radians  to  1.60  radians  at  intervals  of  .01  radian  in  Table  V —  Trigo- 
nometric Functions  in  Badian  Measure.  The  values  of  any  of  these  func- 
tions for  larger  values  of  x  may  be  computed  by  first  converting  the  value 
of  the  angle  in  radian  measure  to  degree  measure,  by  Table  Va,  and  then 
finding  the  value  of  the  function  from  Table  II. 

The  reduction  of  radians  to  degrees  can  be  performed  directly  by  Table 
V  ;  or,  for  greater  accuracy,  by  the  supplementary  Table  Va. 


xviii  EXPLANATION  OF  THE  TABLES  [§  16 

VL     POWERS  —  ROOTS  —  RECIPROCALS 

16.  Arrangement.  This  table  is  arranged  so  that  the  square,  cube, 
square  root,  cube  root,  or  reciprocal  can  be  read  directly  to  five  decimal 
places  for  any  number  n  of  three  significant  figures.  To  attain  this,  not 
only  n2,  n^,  Vn,  Vn,  1/n,  but  also  VlO  n,  VlOn,  VlOO  n  are  printed  on 
every  page.    All  values  have  been  carefully  recomputed  and  checked. 

Thus  to  find  Vl.17,  read  in  V7i  column  the  result:  1.08167.  To  find  Vu.T,  read  in 
the  same  line,  in  's/Ton  column  the  result :  8.42053.  To  find  Vll7,  read  10  times  the 
entry  in  \/n  column,  since  Vll7  =  lOVToY.  

Similarly,  v^I.17  =  1.05373  from  -y/n  column  ;  VH-'^  =  2.27019  from  the  same  line  in 
y/li)  n  column  ;  \/ll7"=  4.89097  from  the  same  line  in  ■yjl'd^n  column. 

The  effect  of  a  change  in  the  decimal  point  in  n^,  n^,  and  1/n  is  only 
to  shift  the  decimal  point  in  the  result,  without  altering  the  digits  printed. 

VIL    NAPIERIAN   OR  NATURAL   LOGARITHMS 

17.  The  Base  e.  —Natural  Logarithms.  The  number  e  =  2.7182818  ... 
is  called  the  natural  base  of  logarithms.  The  logarithms  of  numbers 
to  this  base  are  given  in  Table  VII  at  intervals  of  .01  from  0.01  to 
10.09,  and  at  unit  intervals  from  10  to  409.  The  fundamental  relation 
loge  n  =  loge  10  X  logio  u  cuablcs  us  to  transfer  from  the  base  10  to  the 
base  e,  or  conversely  ;  where  log^  10  =  2.30258509. 

VIIL    MULTIPLES  OF  M  AND   OF  1/M 

18.  Multiples  of  M  and  1/ilf.  This  table  is  convenient  whenever  a 
number  is  to  be  multiplied  by  M  or  by  1/M.  This  occurs  whenever  it  is 
desired  to  change  from  common  logarithms  to  natural  logarithms,  or  con- 
versely, since  M  =  logio  e  and  since  we  have 

logio  X  =  (loge «)  (logio  e)  =  (l/M)\oge  X      and      log,  x  =  M  logio  x. 
Other  formulas  that  require  these  multiples  are 
logio  e^  =  x  logio  e=  X-  M     and      loge(10"  .  x)  =  log^  x  +  n(l/M) ; 
and  the  appropriate  formulas  (see  §§  10,  11,  p.  xiv) 

logio(l  =tx)  =  =tx.if     and      10^^  =  \  ^(l/M)x, 

IX.  VALUES  AND  LOGARITHMS  OF  HYPERBOLIC 
FUNCTIONS 

19.  Hyperbolic  Functions.  This  table  gives  the  values  of  e^,  e-*, 
sinh  aj,  cosh  cc,  tanh  x  ;  and  the  logarithms  of  6^,  sinh  x,  cosh  x,  at  varying 
intervals  from  x  =  0  to  x  =  10.  It  is  to  be  noted  that  log  e-^=  —  log  e* 
and  log  tanh  x  =  log  sinh  x  —  log  cosh  x.  The  table  may  be  extended 
indefinitely  by  means  of  Table  VIII,  since  logio  e*  =  x  .  if ;  for  this  reason 
Table  VIH  may  be  regarded  as  a  table  of  values  of  logio  e^- 


§  22]  VALUES  AND  LOGARITHMS  xix 

X.    VALUES   AND   LOGARITHMS   OF    HAVERSINES 

20.  Haversines.  This  table  gives  the  values  and  the  logarithms  of  the 
haversines  of  angles  from  0°  to  180°  at  intervals  of  10'.  The  haversine, 
which  means  half  of  the  versed  sine,  is 

hav^=(l/2)  vers^=  (1/2)(1  -  cos^)  ; 
hence  its  values  to  five  places  may  be  computed  from  the  table  of  cosines. 
It  is  used  extensively  in  navigation,  and  it  may  be  used  to  advantage  in 
the  solution  of  ordinary  oblique  triangles. 


XL     FACTOR  TABLE  — LOGARITHMS   OF   PRIMES 

21.  Factors  of   Composite  Numbers.    Logarithms  of  Primes.     The 

uses  of  this  table  are  evident  in  questions  involving  factoring,  and  for 
finding  high-place  logarithms  of  numbers  whose  prime  factors  are  less 
than  2018. 

We  shall  illustrate  the  finding  of  logarithms  of  other  numbers  by  finding 
log  TT.  Taking  tt  =  3.14159  26536,  divide  by  3  (the  first  digit),  obtaining 
1.0471975512  •••.  Divide  this  quotient  by  1.047  (in  general,  by  the  nearest 
first  four  digits),  obtaining  1.00018  8683  ....  By  Table  VIII,  the  approxi- 
mate formula  log(l  ±x)  =  ±x  -  M  gives 

log  1.00018  8683  =  .00008  1943  (Table  VIII) 

log  3  =  .47712  12547  (Table  XI) 

log  1.047  =  log  3  -f-  log  .349  =  .01994  66817  (Table  XI) 

logTT  =.497149879 

while  the  true  value  of  log  ir  is  .49714  987269,  so  that  the  error  is  less 
than  1  in  the  eighth  place.  In  general,  this  process  will  give  the  logarithm 
of  any  number  to  within  6  in  the  eighth  decimal  place,  and  the  probable 
error  is  less  than  1.5  in  the  eighth  place.  For  still  greater  accuracy,  see 
Table  la  and  §  10. 

XII.     INTEREST   TABLES 

22.  Interest  Tables.  Tables  XII  a,  6,  c,  d  give  compound  interest 
and  annuity  data  for  various  per  cents  up  to  fifty  years.  Aside  from  the 
obvious  uses,  formulas  involving  this  data  will  be  found  in  works  on 
statistics,  accounting,  and  the  mathematics  of  business. 

Table  XHe  gives  the  logarithms  of  (1  -h  r)  to  fifteen  places,  for  all 
ordinary  values  of  r  from  1/2  ^o  to  10%.  For  other  values  of  r, 
log(l  +  r)  may  be  computed  from  Table  la  (see  §  10).  The  final  result 
in  interest  calculations  may  be  obtained  to  nine  significant  figures  by  the 
antilogarithms  of  Table  la  (see  §  11). 

Table  XII/  is  the  American  Experience  Mortality  Table. 


XX  EXPLANATION   OF  THE   TABLES         [§§23,24 


XIV.     FOUR-PLACE   TABLES 

23-  Four-place  Tables.  These  are  duplicates  of  the  preceding  five- 
place  tables,  reduced  to  four  places,  and  with  larger  intervals  betweer 
the  tabulations.  The  value  of  such  four-place  tables  consists  in  the 
greater  speed  with  which  they  can  be  Used,  in  case  the  degree  of  accuracy 
they  afford  is  sufficient  for  the  purpose  in  hand. 

XlVflf.  Logarithms  of  Numbers.  The  only  special  feature  of  this  table 
is  that  the  proportional  parts  are  printed  for  every  tenth  in  every  row ; 
hence  the  logarithm  of  any  number  of  four  significant  figures  can  be 
read  directly. 

XI V6.  Antilogarithms.  This  table  will  be  found  to  facilitate  approxi- 
mate calculations  to  a  marked  degree.  The  proportional  parts  are  stated 
in  the  right-hand  margin  for  each  row  separately.  This  arrangement, 
with  the  corresponding  one  in  Table  XlVa,  makes  the  tables  effectively 
four-place  each  way. 

XI Vc.  Values  and  Logarithms  of  Trigonometric  Functions.  In  this 
table,  the  values  of  sin  a,  cos  a,  tan  ot,  ctn  a,  and  their  common  loga- 
rithms, are  stated  for  each  10-minute  interval  in  a.  The  characteristics 
of  the  logarithms  are  omitted,  since  they  can  be  supplied  readily  from 
the  value. 


24.  Sources  and  Checks  used.  In  arranging  all  of  these  tables, 
several  extant  tables  have  been  used  as  sources ;  and  the  proofs  have 
been  read  against  the  standard  seven-place  tables  of  Vega,  and  at  least 
one  other  table,  or  against  at  least  two  independent  sources  when  the 
figures  are  not  given  by  Vega.  In  all  cases,  the  stereotyped  plates  have 
been  proof-read  five  times,  by  three  different  persons. 

In  case  of  apparent  doubt,  especially  in  the  last  place  of  decimals,  the 
values  have  been  recomputed,  either  by  series  or  by  the  condensed  fifteen- 
place  tables  of  Hotiel. 

While  errors  may  occur,  it  is  believed  that  they  must  be  purely  typo- 
graphical ;  in  most  cases  such  an  error  is  revealed  by  the  unreasonable 
differences  it  creates. 


Greek 

Alphabet 

Lbttbes  Names 

Letters 

Names 

Letters 

Names 

Lbtters 

Names 

A  a 

Alpha 

H^ 

Eta 

N  V 

Nu 

T    T 

Tau 

B)8 

Beta 

©  0 

Theta 

H^ 

Xi 

Y  V 

Upsilon 

ry 

Gamma 

I    L 

Iota 

O  0 

Omicron 

4>  <^ 

Phi 

AS 

Delta 

K    K 

Kappa 

n  77 

Pi     . 

Xx 

Chi 

E    £ 

Epsilon 

A  X 

Lambda 

Pp 

Rho 

^  il/ 

Psi 

ZC 

Zeta 

M    fJL 

Mu 

S    (T   S 

Sigma 

O  (0 

Omega 

LOGARITHMIC    AND    TRIGOTOMETRIC 
TABLES 


TABLE   I 
COMMON    LOGARITHMS    OF    NUMBERS 


FKOM 

1  TO  10  000 
xo 

FIVE   DECIMAL   PLACES 


1 

-100 

K 

Log 

N 

Log 

N 

Log 

N 

Log 

N 

Log 

0 

20 

1.30  103 

40 

1.60  206 

60 

1.77  815 

80 

1.90  309 

1 
2 

3 

0.00  000 
0.30  103 
0.47  712 

21 
22 
23 

1.32  222 
1.34  242 
1.36  173 

41 
42 
43 

1.61  278 

1.62  325 

1.63  347 

61 
62 
63 

1.78  533 

1.79  239 
1.79  934 

81 
82 
83 

1.90  849 

1.91  381 
1.91  908 

4 

5 
6 

0.60  206 
0.69  897 
0.77  815 

24 
25 

26 

1.38  021 

1.39  794 
1.41  497 

44 
45 

46 

1.64  345 

1.65  321,. 

1.66  276 

64 
65 
66 

1.80  618 

1.81  291 
1.81  954 

84 
85 
86 

1.92  428 

1.92  942 

1.93  450 

7 
8 
9 

10 

0.84  510 
0.90  309 
0.95  424 

27 

28 
29 

1.43  136 

1.44  716 
1.46  240 

47 
48 
49 

1.67  210 

1.68  124 

1.69  020 

67 
68 
69 

1.82  607 

1.83  251 
1.83  885 

87 
88 
89 

1.93  952 

1.94  448 
1.94  939 

1.00  000 

30 

1.47  712 

50 

1.69  897 

70 

1.84  510 

90 

1.95  424 

11 
12 
13 

1.04  139 
1.07  918 
1.11  394 

31 
32 
33 

1.49  136 

1.50  515 

1.51  851 

51 
52 
53 

1.70  757 

1.71  600 

1.72  428 

71 

72 
73 

1.85  126 

1.85  733 

1.86  332 

91 
92 
93 

1.95  90.4 

1.96  379 
1.96  848 

14 

15 
16 

1.14  613 
1.17  609 
1.20  412 

34 
35 
36 

1.53  148 

1.54  407 

1.55  630 

54 
55 
56 

1.73  239 

1.74  036 
1.74  819 

74 
75 
76 

1.86  923 

1.87  506 

1.88  081 

94 
95 
96 

1.9T313 
1.9T772 
1.98  227 

17 
18 
19 

1.23  045 
1.25  527 
1/27  875 

37 
38 
39 

1.56  820 

1.57  978 
1.59  106 

57 
58 
59 

1.75  587 

1.76  343 

1.77  085 

77 
78 
79 

1.88  649 

1.89  209 
1.89  763 

97 

98 
99 

1.98  677 

1.99  123 
1.^)9^564 

N 

Log 

N 

Log 

N 

Log 

N 

Log 

N 

Log 

100  —  Logarithms  of  Numbers  — 160 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

100 

01 
02 
03 

04 
05 
06 

07 

08 
09 

00  000 

043 

087 

130 

173 

217 

260 

303 

346 

389 

432 

860 

01284 

703 

02119 

531 

938 

03  342 

743 

475 

903 
326 

745 
160 
572 

979 

383 

782 

518 
945 
368 

787 
202 
612 

*019 
423 

822 

561 
988 
410 

828 
243 
653 

*060 
463 
862 

604 

*030 

452 

870 
284 
694 

*100 
503 
902 

647 

*072 

494 

912 
325 

735 

*141 
643 
941 

689 

*115 

536 

963 
366 
776 

*181 
683 
981 

732 

*157 

578 

996 
407 
816 

*222 

623 

*021 

776 

*199 

620 

*036 
449 
867 

*262 

663 

*060 

817 

*242 

662 

*078 
490 
898 

*302 

703 

*100 

1 
2 
3 
4 
5 
6 
7 
8 
9 

44 

4.4 
8.8 
13.2 
17.6 
22.0 
26.4 
30.8 
35.2 
39.6 

43 

4.3 
8.6 
12.9 
17.2 
21.5 
25.8 
30.1 
34.4 
38.7 

42 

4.2 
8.4 
12.6 
16.8 
21.0 
25.2 
29.4 
33.6 
37.8 

110 

11 
12 
13 

14 
15 
16 

17 
18 
19 

04139 

179 

218 

258 

297 

336 

376 

415 

454 

493 

1 

532 

922 
05  308 

690 

06070 

446 

819 

07188 
555 

571 
961 
346 

729 
108 
483 

856 
225 
591 

610 
999 
385 

767 
145 
521 

893 
262 
628 

650 

*038 

423 

805 
183 
558 

930 

298 
664 

689 

*077 

461 

843 
221 
595 

967 
335 
700 

727 

*115 

600 

881 
258 
633 

*004 
372 
737 

766 

*164 

638 

918 
296 
670 

*041 

408 
773 

806 

*192 

576 

966 
333 
707 

*078 
446 
809 

844 

*231 

614 

994 
371 
744 

*115 

482 
846 

883 

*269 

652 

*032 
408 

781 

*151 
618 

882 

1 
2 
3 
4 
5 
6 
7 
8 
9 

41 

4.1 
8.2 
12.3 
16.4 
20.6 
24.6 
28.7 
32.8 
36.9 

40 

4.0 
8.0 
12.0 
16.0 
20.0 
24.0 
28.0 
32.0 
36.0 

39 

3.9 
7.8 
11.7 
15.6 
19.5 
23.4 
27.3 
31.2 
35.1 

120 

918 

954 

990 

*027 

*063 

*099 

*135 

*171 

*207 

*243 

1 

21 
22 
23 

24 
25 
26 

27 
28 
29 

08279 
636 
991 

09342 

691 

10037 

380 

721 

11059 

314 

672 

*026 

377 
726 
072 

415 
755 
093 

350 

707 

*061 

412 
760 
106 

449 
789 
126 

386 

743 

*096 

447 
795 
140 

483 
823 
160 

422 

778 
*132 

482 
830 
175 

517 
857 
193 

458 

814 

*167 

517 
864 
209 

551 
890 
227 

493 

849 

*202 

562 
899 
243 

685 
924 
261 

529 

884 
*237 

587 
934 

278 

619 

968 
294 

665 
920 

*272 

621 
968 
312 

653 
992 
327 

600 

965 

*307 

656 

*003 

346 

687 

*026 

361 

1 
2 
3 

4 
5 
6 
7 
8 
9 

38 

3.8 
7.6 
11.4 
16.2 
19.0 
22.8 
26.6 
30.4 
34.2 

37 

3.7 
7.4 
11.1 
14.8 
18.5 
22.2 
26.9 
29.6 
33.3 

36 

3.6 
7.2 
10.8 
14.4 
18.0 
21.6 
25.2 
28.8 
32.4 

130 

394 

428 

461 

494 

528 

561 

694 

628 

661 

694 

1 

31 
32 
33 

34 
35 
36 

37 
38 
39 

727 

12057 

385 

710 

13033 

354 

672 

988 

14301 

760 
090 
418 

743 
066 
386 

704 

*019 

333 

793 
123 
450 

775 
098 
418 

735 

*051 

364 

826 
156 
483 

808 
130 
450 

767 

*082 

395 

860 
189 
516 

840 
162 
481 

799 

*114 

426 

893 
222 
548 

872 
194 
513 

830 

*145 

457 

926 
264 
681 

905 
226 
646 

862 

*176 

489 

959 

287 
613 

937 

258 

577 

893 

*208 

520 

992 
320 
646 

969 
290 
609 

926 

*239 
551 

*024 
352 
678 

*001 
322 
640 

956 

*270 
682 

891 

1 
2 
3 
4 
6 
6 
7 
8 
9 

35 

3.6 
7.0 
10.6 
14.0 
17.6 
21.0 
24.5 
28.0 
31.5 

34 

3.4 
6.8 
10.2 
13.6 
17.0 
20.4 
23.8 
27.2 
30.6 

33 

3.3 
6.6 
9.9 
13.2 
16.5 
19.8 
23.1 
26.4 
29.7 

140 

613 

644 

675 

706 

737 

768 

799 

829 

860 

1 

41 
42 
43 

44 
45 
46 

47 
48 
49 

922 

15229 

534 

836 

16137 

435 

732 

17026 

319 

953 
259 
564 

866 
167 
465 

761 
056 
348 

983 
290 
594 

897 
197 
495 

791 
085 
377 

*014 
320 
625 

927 
227 
524 

820 
114 
406 

*045 
351 
655 

957 
256 
654 

850 
143 
435 

*076 
381 
685 

987 
286 
684 

879 
173 
464 

*106 
412 
716 

*017 
316 
613 

909 
202 
493 

*137 
442 
746 

*047 
346 
643 

938 
231 
522 

*168 
473 
776 

*077 
376 
673 

967 
260 
661 

*198 
503 
806 

*107 
406 
702 

997 

289 
680 

1 

2 
3 
4 
6 
6 
7 
8 
9 

32 

3.2 
6.4 
9.6 
12.8 
16.0 
19.2 
22.4 
25.6 
28.8 

31 

3.1 

6.2 
9.3 
12.4 
15.5 
18.6 
21.7 
24.8 
27.9 

30 

3.0 
6.0 
9.0 
12.0 
15.0 
18.0 
21.0 
24.0 
27.0 

150 

609 

638 

667 

696 

725 

754 

782 

811 

840 

869 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

11 

150- 

-  Logarithms  of  Numbers 

—  200 

3 

N. 

0 

1    2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

150 

17  609 

638 

667 

696 

725 

754 

782 

811 

840 

869 

51 

898 

926 

955 

984 

*013 

*041 

*070 

*099 

*127 

*156 

52 

18184 

213 

241 

270 

298 

327 

355 

384 

412 

441 

53 

469 

498 

526 

554 

583 

611 

639 

667 

696 

724 

54 

752 

780 

808 

837 

^65 

893 

921 

949 

977 

*005 

55 

19  033 

061 

089 

117 

145 

173 

201 

229 

257 

285 

56 

312 

340 

368 

396 

424 

451 

479 

507 

535 

562 

57 

590 

618 

645 

673 

700 

728 

756 

783 

811 

838 

58 

866 

893 

921 

948 

976 

*003 

*030 

*058 

*085 

*112 

59 

20140 

167 

194 

222 

249 

276 

303 

330- 

358 

385 

160 

412 

439 

466 

493 

620 

548 

575, 

602 

629 

656 

61 

683 

710 

737 

763 

790 

817 

844 

871 

898 

925 

29 

28 

27 

62 

952 

978 

*005 

*032 

*059 

*085 

*112 

*139 

*165 

*192 

1 

2.9 

2.8 

2.7 

63 

21219 

245 

272 

299 

325 

352 

378 

405 

431 

458 

2 

5.8 

5.6 

5.4 

64 

484 

511 

537 

564 

590 

617 

643 

669 

696 

722 

3 

8.7 

8.4 

8.1 

65 

748 

775 

801 

827 

854 

880 

906 

932 

958 

985 

4 

11.6 

11.2 

10.8 

m 

22  011 

037 

063 

089 

115 

141 

167 

194 

220 

246 

5 

14.5 
17.4 

14.0 
16.8 

13.5 
16.2 

67 

272 

298 

324 

350 

376 

401 

427 

453 

479 

505 

7 

20.3 

19.6 

18.9 

68 

531 

557 

583 

608 

634 

660 

686 

712 

737 

763 

8 

23.2 

22.4 

21.6 

69 

789 

814 

840 

866 

891 

917 

943 

968 

994 

*019 

9 

26.1 

25.2 

24.3 

170 

23045 

070 

096 

121 

147 

172 

198 

223 

249 

274 

1 

71 

300 

325 

350 

376 

401 

426 

452 

477 

502 

528 

26 

25 

24 

72 

553 

578 

603 

629 

654 

679 

704 

729 

754 

779 

1 

2.6 

2.5 
5.0 

2.4 

73 

805 

830 

855 

880 

905 

930 

955 

980 

*005 

*030 

2 

5.2 

4.'8 

74 

24  055 

080 

105 

130 

155 

180 

204 

229 

254 

279 

^ 

7.8 

7.5 

7.2 

75 

304 

329 

353 

378 

403 
650 

428 

452 

477 

502 

527 

4 

10.4 

10.0 

9.6 

76 

551 

576 

601 

625 

674 

699 

724 

748 

773 

5 

13.0 

12.5 

12.0 

6 

15.6 

15.0 

14.4 

77 

797 

822 

846 

871 

895 

920. 

944 

969 

993 

*018 

7 

18.2 

17.5 

16.8 

78 

25  042 

066 

091 

115 

139 

164 

188 

212 

237 

261 

8 

20.8 

20.0 

19.2 

79 

285 

310 

334 

358 

382 

406 

431 

455 

479 

503 

9 

23.4 

22.5 

21.6 

180 

527 

551 

575 

600 

624 

648 

672 

696 

720 

744 

1 

81 

768 

792 

816 

840 

864 

888 

912 

935 

959 

983 

23 

22 

21 

82 

26  007 

031 

055 

079 

102 

126 

150 

174 

198 

221 

1 
2 

2.3 
4.6 

2.2 
4.4 

2.1 
4.2 

83 

245 

269 

293 

316 

340 

364 

387 

411 

435 

458 

84 

482 

505 

529 

553 

576 

600 

623 

647 

670 

694 

3 

6.9 

6.6 

6.3 

85 

717 

741 

764 

788 

811 

834 

858 

881 

905 

928 

4 

9.2 

8.8 

8.4 

86 

951 

975 

998 

*021 

*045 

*068 

*091 

*114 

*138 

*161 

5 
6 

11.5 
13.8 

11.0 
13.2 

10.5 
12.6 

87 

27184 

207 

231 

254 

277 

300 

323 

346 

370 

393 

7 

16.1 

15.4 

14.7 

88 

416 

439 

462 

485 

508 

531 

554 

577 

600 

623 

8 

18.4 

17.6 

16.8 

89 

646 

669 

692 

715 

,738 

761 

784 

807 

830 

852 

9 

20.7 

19.8 

18.9 

190 

875 

898 

921 

944 

967 

989 

*012 

*035 

*058 

*081 

91 

28103 

126 

149 

171 

194 

217 

240 

262 

285 

307 

92 

330 

353 

375 

398 

421 

443 

466 

488 

511 

533 

93 

556 

57« 

601 

623 

646 

668 

691 

713 

735 

758 

94 

780 

803 

825 

847 

870 

892 

914 

937 

959 

981 

95 

29003 

026 

048 

070 

092 

115 

137 

159 

181 

203 

96 

226 

248 

270 

292 

314 

336 

358 

380 

403 

425 

97 

447 

469 

491 

513 

535 

557 

579 

601 

623 

645 

98 

667 

688 

710 

732 

754 

776 

798 

820 

842 

863 

99 

885 

907 

929 

951 

973 

994 

*016 

*038 

*060 

*081 

200 

30103 

125 

146 

168 

190 

211 

233 

255 

276 

298 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8   9 

Prop.  Pts.    1 

200  —  Logarithms  of  Numbers  —  250 


N. 


8    9 


Prop.  Pts. 


200 


210 


220 


230 

31 
32 
33 

34 
35 
36 

37 
38 
39 


240 


41 
42 
43 

44 
45 
46 

47 
48 
49 

250 


30103 


125 


146 


168 


190 


211 


233 


255 


276 


298 


320 
535 
750 

963 
31175 

387 

597 

806 

32  015 


341 
557 
771 

984 
197 
408 

618 
827 
035 


363 
578 
792 

*006 
218 
429 

639 

848 
056 


384 
600 
814 

*027 
239 
450 

660 
869 

077 


406 
621 
835 

*048 
260 
471 

681 
890 
098 


428 
643 
856 

*069 
281 
492 

702 
911 
118 


449 

664 
878 

*091 
302 
613 

723 

931 
139 


471 

685 
899 

*112 
323 
534 

744 
952 
160 


492 
707 
920 

*133 
345 
555 

765 
973 
181 


514 

728 
942 

*154 
366 
576 

785 
994 
201 


222 


243 


263 


284 


305 


346 


366 


387 


408 


428 
634 

838 

33041 
244 
445 

646 

846 

34044 


449 

654 
858 

062 
264 
465 

666 
866 
064 


469 
675 
879 

082 

284 
486 

686 

885 
084 


490 

695 
899 

102 
304 
506 

706 
905 
104 


510 
715 
919 

122 
325 
526 

726 
925 
124 


531 
736 
940 

143 

345 
546 

746 
945 
143 


552 
756 
960 

163 
365 
666 

766 
965 
163 


572 
777 
980 

183 

385 
586 

786 
985 
183 


593 

797 

*001 

203 
405 
606 

806 

*005 

203 


613 

818 
*021 

224 
425 
626 

826 

*025 

223 


242 


262 


282 


301 


321 


341 


361 


380 


400 


420 


439 
635 
830 

35025 

218 
411 

603 
793 
984 


459 
655 

850 

044 

238 
430 

622 

813 

*003 


479 
674 
869 

064 
257 
449 

641 

832 

*021 


498 
694 
889 

083 
276 
468 

660 

851 

*040 


518 
713 
908 

102 

295 
488 

679 

870 

*059 


537 
733 
928 

122 

315 
507 

698 

889 

*078 


557 
753 

947 

141 
334 
526 

717 

908 
*097 


677 
772 
967 

im 

353 
545 

736 

927 

*116 


596 

792 
986 

180 
372 
564 

755 

946 

*135 


616 

811 

*005 

199 
392 
583 

774 

965 

*154 


36173 


192 


211 


229 


248 


267' 


286 


305 


324 


342 


361 
549 
736 

922 

37107 

291 

475 

658 
840 


380 

568 
754 

940 
125 
310 

493 

676 
858 


399 
586 
773 

959 
144 
328 

511 

694 
876 


418 
605 
791 

977 
162 
346 

530 
712 

894 


436 
624 
810 

996 
181 
365 

548 
731 
912 


455 
642 
829 

*014 
199 
383 

566 
749 
931 


474 
661 
847 

*033 
218 
401 

585 
767 
949 


493 
680 
866 

*051 
236 
420 

603 

785 
967 


511 

698 
884 

*070 
254 
438 

621 

803 
985 


530 
717 
903 

*088 
273 
457 

639 

822 
*003 


38021 


039 


057 


075 


093 


112 


130 


148 


166 


184 


202 
382 
561 

739 

917 

39094 

270 
445 
620 

794 


220 
399 
578 

757 
934 
111 

287 
463 
637 

811 


238 
417 
596 

775 
952 
129 

305 
480 
655 

829 


256 
435 
614 

792 
970 
146 

322 
498 
672 

846 


274 
453 

632 

810 
987 
164 

340 
515 
690 

863 


292 
471 
650 

828 
*005 

182 

358 
533 
707 

881 


310 

489 
668 

846 

*023 

199 

375 
550 

724 

898 


328 
507 
686 

863 

*041 

217 

393 

568 
742 

915 


346 
525 
703 

881 

*058 

235 

410 

585 
759 

933 


364 
543 
721 


*076 
252 

428 
602 

777 

950 


log  2  =.30102  99566 


22 

21 

2.2 

2.1 

4.4 

4.2 

G.a 

6.3 

8.8 

8.4 

11.0 

10.5 

13.2 

12.6 
14.7 

15.4 

17.6 

16.8 

19.8 

18.9 

19 

18 

1.9 

1.8 

3.8 

3.6 

5.7 

5.4 

7.6 

7.2 

9.5 

9.0 

11.4 

10.8 

13.3 

12.6 

15.2 

14.4 

17.1 

16.2 

Prop.  Pts. 


I] 

250- 

-  Logarithms  of  Numbers 

»  — 300 

5 

N. 

0 

1 

2 

3 

4 

3 

6 

7 

8 

9 

Prop.  Pts. 

250 

39  794 

811 

829 

846 

863 

881 

898 

915 

933 

950 

51 

967 

985 

*002 

*019 

*037 

*054 

*071 

*088 

*106 

*123 

52 

40140 

157 

175 

192 

209 

226 

243 

261 

278 

295 

53 

312 

329 

346 

364 

381 

398 

415 

432 

449 

466 

54 

483 

500 

518 

635 

552 

569 

586 

603 

620 

637 

55 

654 

671 

688 

705 

722 

739 

756 

773 

790 

807 

56 

824 

841 

858 

875 

892 

909 

926 

943 

960 

976 

57 

993 

*010 

*027 

*044 

*061 

*078 

*095 

nil 

*128 

*145 

58 

41162 

179 

196 

212 

229 

246 

263 

280 

296 

313 

59 

330 

347 

363 

380 

397 

414 

430 

447 

464 

481 

260 

497 

514 

531 

547 

5(54 

581 

597 

614 

631 

647 

61 

664 

681 

697 

714 

731 

747 

764 

780 

797 

814 

18 

17 

16 

62 

830 

847 

863 

880 

896 

913 

929 

946 

963 

979 

1 

1.8 

1.7 

1.6 

63 

996 

*012 

*029 

*045 

*062 

*078 

*095 

nil 

*127 

*144 

2 

3.6 

3.4 

3.2 

64 

42160 

177 

193 

210 

226 

243 

259 

275 

292 

308 

3 
4 
5 
6 

5.4 

7.2 

9.0 

10.8 

6.1 

6.8 

8.5 

10.2 

4.8 
6.4 
8.0 
9.6 

65 

325 

Ml 

357 

374 

390 

406 

423 

439 

455 

472 

66 

488 

504 

521 

537 

553 

570 

586 

602 

619 

635 

67 

651 

667 

684 

700 

716 

732 

749 

765 

781 

797 

7 

12.6 

11.9 

11.2 

68 

813 

830 

846 

862 

878 

894 

911 

927 

943 

959 

8 

14.4 

13.6 

12.8 

69 

975 

991 

*008 

*024 

*040 

*056 

*072 

*088 

*104 

*120 

9 

16.2 

15.3 

14.4 

270 

43136 

152 

169 

185 

201 

217 

233 

249 

265 

281 

71 

297 

313 

329 

345 

361 

377 

393 

409 

425 

441 

72 

457 

473 

489 

505 

521 

537 

553 

569 

584 

600 

73 

616 

632 

648 

664 

680 

696 

712 

727 

743 

759 

M=\ogioe 

74 

775 

791 

807 

823 

838 

854 

870 

886 

902 

917 

=  logio2.718... 
=  .4342944819 

75 

933 

949 

965 

981 

996 

*012 

*028 

*044 

*059 

*075 

76 

44091 

107 

122 

138 

154 

170 

185 

201 

217 

232 

77 

248 

264 

279 

295 

311 

326 

342 

358 

373 

389 

78 

404 

420 

436 

451 

467 

483 

498 

514 

529 

545 

79 

560 

576 

592 

607 

623 

638 

654 

669 

685 

700 

280 

716 

731 

747 

762 

778 

793 

809 

824 

840 

855 

81 

871 

886 

902 

917 

932 

948 

963 

979 

994 

*010 

15 

14 

82 

45  025 

040 

056 

071 

086 

102 

117 

133 

148 

163 

1 
2 

1.5 
3.0 

1.4 

2.8 

83 

179 

194 

209 

225 

240 

255 

271 

286 

301 

317 

84 

332 

347 

362 

378 

393 

408 

423 

439 

454 

469 

3 

4.5 

4.2 

85 

484 

500 

515 

530 

545 

561 

576 

591 

606 

621 

4 

6.0 

5.6 

86 

637 

652 

667 

682 

697 

712 

728 

743 

758 

773 

5 
6 

7 

7.5 

9.0 

10.5 

7.0 
8.4 
9.8 

87 

788 

803 

818 

834 

849 

864 

879 

894 

909 

924 

88 

939 

954 

969 

984 

*000 

*015 

*030 

*045 

*060 

*075 

8 

12.0 

11.2 

89 

46  090 

105 

120 

135 

150 

165 

180 

195 

210 

225 

9 

13.5 

12.6 

290 

240 

255 

270 

285 

300 

315 

330 

345 

359 

374 

91 

389 

404 

419 

434 

449 

464 

479 

494 

509 

523 

92 

538 

553 

568 

583 

598 

613 

627 

642 

657 

672 

93 

687 

702 

716 

731 

746 

-761 

776 

790 

805 

820 

94 

835 

850 

8(54 

879 

894 

909 

923 

938 

953 

967. 

95 

982 

997 

*012 

*026 

*041 

*056 

*070 

*085 

*100 

ni4 

96 

47129 

144 

159 

173 

188 

202 

217 

232 

246 

261 

97 

276 

290 

305 

319 

334 

349 

363 

378 

392 

407 

98 

422 

436 

451 

465 

480 

494 

509 

524 

538 

553 

_99 

567 

582 

596 

611 

625 

640 

654 

669 

683 

698 

300 

712 

727 

741 

756 

770 

784 

799 

813 

828 

842 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

6 

300- 

-  Logarithms 

of  Numbers 

-350 

D 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

300 

47  712 

727 

741 

756 

770 

784 

799 

813 

828 

842 

01 
02 
03 

857 

48  001 

144 

871 
015 
159 

885 
029 
173 

900 
044 

187 

914 
058 
202 

929 
073 
216 

943 

087 
230 

958 
101 
244 

972 
116 
259 

986 
130 
273 

04 
05 
06 

287 
430 
572 

302 
444 

586 

316 
458 
601 

330 
473 
615 

344 

487 
629 

359 
501 
643 

373 
615 
657 

387 
530 
671 

401 
544 
686 

416 
558 
700 

log  3  =.47712 12547 
log  77=  .4971498727 

07 

.  08 

09 

714 

855 
996 

728 

869 

*010 

742 

883 

*024 

756 

897 

*038 

770 
911 

*052 

785 

926 

*066 

799 

940 

*080 

813 

954 

*094 

827 

968 

*108 

841 

982 
*122 

310 

49136 

150 

164 

178 

192 

206 

220 

234 

248 

262 

11 
12 
13 

276 
415 
554 

290 
429 
568 

304 
443 

582 

318 
457 
596 

332 
471 
610 

346 
485 
624 

360 
499 
638 

374 
613 
651 

388 
527 
665 

402 
541 
679 

1 
2 
3 
4 
6 
6 

10 

1.5 
3.0 
4.5 
6.0 
7.5 
9.0 

14 

1.4 

2.8 
4.2 
5.6 
7.0 
8.4 

14 
15 
16 

693 
831 
969 

707 
845 

982 

721 

859 
996 

734 

872 
*010 

748 

886 

*024 

762 

900 
*037 

776 

914 

*051 

790 

927 

*065 

803 

941 

*079 

817 

955 

*092 

17 
18 
19 

50106 
243 

379 

120 
256 
393 

133 
270 
406 

147 
284 
420 

161 

297 
433 

174 
311 
447 

188 
325 
461 

202 
338 
474 

215 
352 

488 

229 

365 
501 

7 
8 
9 

10.5 
12.0 
13.5 

9.8 
11.2 
12.6 

320 

515 

529 

542 

556 

569 

583 

596 

610 

623 

637 

21 

22 
23 

651 

786 
920 

664 
799 
934 

678 
813 
947 

691 
826 
961 

705 
840 
974 

718 
853 
987 

732 

866 
*001 

745 

880 

*014 

759 

893 

*028 

772 

907 

*041 

24 
25 
26 

51055 
188 
322 

068 
202 
335 

081 
215 
348 

095 
228 
362 

108 
242 
375 

121 

255 

388 

135 

268 
402 

148 
282 
415 

162 
295 

428 

175 
308 
441 

27 

28 
29 

455 

587 
720 

468 
601 
733 

481 
614 
746 

495 
627 

759 

508 
640 

772 

521 
654 

786 

534 

667 
799 

648 
680 
812 

561 
693 

825 

574' 
706 
838 

330 

851 

865 

878 

891 

904 

917 

930 

943 

957 

970 

31 
32 
33 

983 

52114 

244 

996 
127 
257 

*009 
140 
270 

*022 
153 

284 

*035 
166 
297 

*048 
179 
310 

*061 
192 
323 

*075 
205 
336 

*088 
218 
349 

*101 
231 
362 

1 
2 

13 

1.3 

2.6 

12 

1.2 
2.4 
3.6 

4.8 
6.0 

7.2 

34 
35 
36 

375 
504 
634 

388 
517 
647 

401 
530 
660 

414 
543 

673 

427 
556 
686 

440 
569 
699 

453 
582 
711 

466 
595 
724 

479 
608 
737 

492. 
621 
750 

3 
4 
6 
6 

3.9 
5.2 
6.5 

7.8 

37 
38 
39 

763 

892 

53020 

776 
905 
033 

789 
917 
046 

802 
930 
058 

815 
943 
071 

827 
956 
084 

840 
969 
097 

853 
982 
110 

866 
994 
122 

879 

*007 

135 

7 
8 
9 

9.1 
10.4 
11.7 

8.4 

9.6 

10.8 

340 

148 

161 

173 

186 

199 

212 

224 

237 

250 

263 

41 
42 
43 

275 
403 
529 

288 
415 
542 

301 

428 
555 

314 
441 
567 

326 
453 

580 

339 
4(56 
593 

352 
479 
605 

364 
491 
618 

377 
504 
631 

390 
517 
643 

44 
45 
46 

656 
782 
908 

668 
794 
920 

681 
807 
933 

694 
820 
945 

706 
832 
958 

719 
845 
970 

732 

857 
983 

744 
870 
995 

757 

882 

*008 

769 

895 

*020 

47 
48 
49 

54033 
158 
283 

045 
170 
295 

058 
183 
307 

070 
195 
320 

083 
208 
332 

095 
220 
345 

108 
233 
357 

120 
245 

370 

133 

258 
382 

145 
270 

394 

350 

407 

419 

432 

444 

456 

469 

481 

494 

506 

518 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

I] 

350- 

-Logarithms  of  Numbers  —  400 

7 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

350 

54  407 

419 

432 

444 

456 

469 

481 

494 

506 

518 

51 

,52 
53 

531 
654 

777 

543 
667 
790 

555 
679 

802 

568 
691 
814 

580 
704 

827 

593 

716 
839 

605 
728 
851 

617 
741 

864 

630 

753 
876 

(542 
765 
888 

54 
55 
56 

900 

55  023 

145 

913 
035 
157 

925 
047 
169 

937 
060 

182 

949 
072 
194 

962 
084 
206 

974 

096 
218 

986 
108 
230 

998 
121 
242 

*011 
133 
255 

57 
58 
59 

360 

267 
388 
509 

279 
400 
522 

291 
413 
534 

303 
425 
546 

315 
437 

558 

328 
449 
570 

340 
461 

582 

352 
473 
594 

364 
485 
606 

376 

497 
618 

630 

642 

654 

6m 

678 

691 

703 

715 

727 

739 

(51 
()2 
(53 

751 
871 
991 

763 

883 
*003 

775 

895 

*015 

787 

907 

*027 

799 

919 

*038 

811 

931 

*050 

823 

943 

*062 

835 

955 

*074 

847 

967 

*086 

859 
979 

*098 

1 
2 

13 

1.3 
2.6 

12 

1.2 
2.4 

(34 
(55 

m 

56110 
229 
348 

122 
241 

360 

134 
253 

372 

146 

265 

384 

158 

277 
396 

170 

289 
407 

182 
301 
419 

194 
312 
431 

205 
324 
443 

217 
336 
455 

3 
4 
5 
6 

3.9 
5.2 
6.5 

7  8 

3.6 
4.8 
6.0 
7  2 

(57 
(58 
69 

467 
585 
703 

478 
597 
714 

490 
608 
726 

502 
620 
738 

514 
632 

750 

526 
644 
761 

538 
656 
773 

549 
667 

785 

561 
679 
797 

573 
691 
808 

7 
8 
9 

9.1 
10.4 
11.7 

8.4 

9.6 

10.8 

370 

820 

832 

844 

855 

867 

879 

891 

902 

914 

926 

71 

72 
73 

937 

57  054 

171 

949 
066 
183 

961 
078 
194 

972 

089 
206 

984 
101 
217 

9fX) 
113 
229 

*008 
124 
241 

*019 
136 
252 

*031 
148 
264 

*043 
159 
276 

74 
75 
76 

287 
403 
519 

299 
415 
530 

310 
426 
542 

322 
438 
553 

334 
449 
565 

345 
4(51 
576 

357 
473 
588 

368 
484 
600 

380 
4% 
611 

392 
507 
623 

77 
78 
79 

634 

749 
864 

646 
761 

875 

657 

772 
887 

669 
784 
898 

680 
795 
910 

692 
807 
921 

703 

818 
933 

715 
830 
944 

726 
841 
955 

738 
852 
967 

380 

978 

990 

*001 

*013 

*024 

*035 

*047 

*058 

*070 

*081 

81 
82 
83 

58  092 
206 
320 

104 
218 
331 

115 
229 
343 

127 
240 
354 

138 
252 
365 

149 
263 
377 

161 
274 

388 

172 

286 
399 

184 
297 
410 

195 
309 
422 

1 
2 

11 

1.1 

10 

1.0 
2.0 

84 

85 
86 

433 
546 
659 

444 
557 
670 

456 
569 
681 

467 
580 
692 

478 
591 
704 

490 
602 
715 

501 
614 
726 

512 
625 
737 

524 
636 
749 

535 

647 
760 

3 
4 
5 
6 

7 
8 
9 

3.3 
4.4 
5.5 
6.6 

7.7 
8.8 
9.9 

3.0 
4.0 
5.0 
6.0 
7.0 
8.0 
9.0 

87 
88 
89 

771 

883 
995 

782 

894 

*006 

794 

906 

*017 

805 
917 

*028 

816 

928 

*040 

827 

939 

*051 

838 

950 

*062 

850 

961 

*073 

861 
973 

*084 

872 

984 

*095 

390 

59106 

118 

129 

140 

151 

162 

173 

184 

195 

207 

91 
92 
93 

218 
329 
439 

229 
340 
450 

240 
351 
461 

251 
362 
472 

262 
373 

483 

273 
384 
494 

284 
395 
506 

295 
406 
517 

306 
417 
528 

318 
428 
539 

94 
95 
96 

550 
660 

770 

561 
671 

780 

572 
682 
791 

583 
693 

802 

594 
704 
813 

605 
715 

824 

616 
726 

835 

627 
737 
846 

638 
748 
857 

649 
759 

868 

,97 
98 
99 

879 

988 

60  097 

890 
999 
108 

901 

*010 

119 

912 

*021 

130 

923 

*032 

141 

934 

*043 

152 

945 

*054 

163 

956 

*065 

173 

966 

*076 

184 

977 

*086 

195 

400 

206 

217 

228 

239 

249 

260 

271 

282 

293 

304 

^. 

0 

1 

2 

3 

4 

5 

6 

7    8 

9 

Prop.  Pts. 

400  —  Logarithms  of  Numbers  —  450 


N. 

0 

1 

2 

3 

4 

6 

6 

7 

8 

9 

Prop.  Pts. 

400 

60  206 

217 

228 

239 

249 

260 

271 

282 

293 

304 

01 
02 
03 

04 
05 
06 

07 

08 
09 

314 
423 
531 

638 
746 
853 

959 

61066 

172 

325 
433 
541 

649 
756 

863 

970 
077 
183 

336 
444 
552 

660 

767 

874 

981 

087 
194 

347 
455 
563 

670 

778 
885 

991 

098 
204 

358 
466 
574 

681 

788 
895 

*002 
109 
215 

369 

477 
584 

692 

799 
906 

*013 
119 
225 

379 

487 
595 

703 
810 
917 

*023 
130 
236 

390 
498 
606 

713 
821 
927 

*034 
140 
247 

401 
509 
617 

724 
831 
938 

*045 
151 

257 

412 
520 
627 

735 
842 
949 

*055 
162 

268 

410 

278 

289 

300 

310 

321 

331 

342 

352 

363 

374 

11 
12 
13 

14 
15 
16 

17 
18 
19 

384 
490 
595 

700 
805 
909 

62014 
118 
221 

395 
500 
606 

711 

815 
920 

024 
128 
232 

405 
511 
616 

721 
826 
930 

034 
138 
242 

416 
521 
627 

731 
836 
941 

045 
149 
252 

426 
532 
637 

742 
847 
951 

055 
159 
263 

437 
542 
648 

752 

857 
962 

066 
170 
273 

448 
553 
658 

763 
868 
972 

076 
180 

284 

458 
563 
669 

773 
878 
982 

086 
190 
294 

469 
574 
679 

784 
888 
993 

097 
201 
304 

479 

584 
690 

794 

899 
*003 

107 
211 
315 

420 

325 

335 

346 

356 

366 

377 

387 

397 

408 

418 

21 
22 
23 

24 
25 
26 

27 
28 
29 

428 
531 
634 

737 
839 
941 

63043 
144 
246 

439 
542 
644 

747 
849 
951 

053 
155 

256 

449 
552 
655 

757 
859 
961 

063 
165 
266 

459 
562 
665 

767 
870 
972 

073 
175 

276 

469 
572 
675 

778 
880 
982 

083 
185 

28(3 

480 
583 
685 

788 

992 

094 
195 

296 

490 
593 
696 

798 

900 

*002 

104 
205 
306 

500 
603 
706 

808 

910 

*012 

114 
215 
317 

511 
613 
716 

818 

921 

*022 

124 
225 
327 

521 
624 
726 

829 

931 

*033 

134 

236 
337 

1 
2 

3 
4 
5 
6 

7 
8 
9 

11 

1.1 
2.2 
3.3 
4.4 
5.5 
6.6 
7.7 
8.8 
9.9 

10 

1.0 
2.0 
3.0 
4.0 
5.0 
6.0 
7.0 
8.0 
9.0 

9 

0.9 
1.8 
2.7 
3.6 
4.5 
5.4 
6.3 
7.2 
8.1 

430 

347 

357 

367 

377 

387 

397 

407 

417 

428 

438 

lo^if=log[loge] 
=  9.63778431  —  10 

31 
32 
33 

34 
35 
36 

37 
38 
39 

448 
548 
649 

749 
849 
949 

64048 
147 
246 

458 
558 
659 

759 
859 
959 

058 
157 
256 

468 
568 
669 

769 
869 
969 

068 
167 
266 

478 
579 
679 

779 
879 
979 

078 
177 
276 

488 
589 
689 

789 
889 
988 

088 

187 
286 

498 
599 
699 

799 
899 
998 

098 
197 
296 

508 
609 
709 

809 

909 

*008 

108 
207 
306 

518 
619 
719 

819 

919 

*018 

118 
217 
316 

528 
629 
729 

829 

929 

*028 

128 
227 
326 

538 
639 
739 

839 
939 

*038 

137 
237 
335 

440 

345 

355 

365 

375 

385 

395 

404 

414 

424 

434 

41 
42 
43 

44 
45 
46 

47 
48 
49 

444 
542 
640 

738 
836 
933 

65031 

128 
225 

454 
552 
650 

748 
846 
943 

040 
137 
234 

464 
562 
660 

758 
856 
953 

050 
147 
244 

473 
572 
670 

768 
865 
963 

060 
157 
254 

483 
582 
680 

777 
875 
972 

070 
167 
263 

493 
591 

689 

787 
885 
982 

079 
176 

273 

503 
601 
699 

797 
895 
992 

089 
186 

283 

513 
611 
709 

807 

904 

*002 

099 
196 

292 

523 
621 
719 

816 
914 

*ou 

108 
205 
302 

532 
631 
729 

826 

924 

*021 

118 
215 
312 

450 

321 

331 

341 

350 

360 

369 

379 

389 

398 

408 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

u 

450- 

-  Logarithms 

Of  Numbers  — 500 

9 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

450 

65  321 

331 

341 

;350 

360 

3()9 

379 

389 

398 

408 

51 
52 
53 

418 
514 
610 

427 
523 
619 

437 
533 
629 

447 
543 
639 

456 
552 
648 

466 
562 
658 

475 
571 
667 

485 
581 
677 

495 
591 
686 

504 
600 
696 

54 
55 
56 

706 
801 
896 

715 

811 
906 

725 
820 
916 

734 
830 
925 

744 
839 
935 

753 
849 
944 

763 
858 
954 

772 
868 
963 

782 
877 
973 

792 
887 
982 

57 
58 
59 

992 

66  087 

181 

*001 
096 
191 

*011 
106 
200 

*020 
115 
210 

*030 
124 
219 

*039 
134 
229 

*049 
143 

238 

*058 
153 
247 

*068 
162 
257 

*077 
172 
266 

460 

276 

285 

295 

304 

314 

323 

332 

342 

351 

361 

61 
62 
63 

370 
464 

558 

380 
474 
567 

389 
483 

577 

398 
492 
586 

408 
502 
596 

417 
511 
605 

427 
521 
614 

436 
530 
624 

445 
539 
633 

455 
549 
642 

64 
65 

m 

652 
745 
839 

661 
755 
848 

671 
764 

857 

680 
773 
867 

689 

783 
876 

699 
792 
885 

708 
801 
894 

717 
811 
904 

727 
820 
913 

736 
829 
922 

67 
68 
69 

932 

67  025 

117 

941 
034 
127 

950 
043 
136 

960 
052 
145 

969 
062 
154 

978 
071 
164 

987 
080 
173 

997 
089 

182 

*006 
191 

*015 
108 
201 

470 

210 

219 

228 

237 

247 

256 

265 

274 

284 

293 

71 

72 
73 

302 
394 

486 

311 
403 
495 

321 
413 
504 

330 
422 
514 

339 
431 
523 

348 
440 
532 

357 
449 
541 

367 
459 
550 

376 
468 
560 

385 
477 
569 

1 
2 

10 

1.0 
2.0 

9 

0.9 
1.8 

8 

0.8 
1.6 

74 
75 
76 

578 
669 
761 

587 
679 
770 

596 

688 
779 

605 

697 
788 

614 

706 
797 

624 
715 
806 

633 
724 

815 

642 

733 

825 

651 

742 
834 

660 

7r)2 

843 

3 
4 
5 
6 

3.0 
4.0 
5.0 
60 

2.7 
3.6 
4.5 
5  4 

2.4 
3.2 
4.0 

4.8 

77 
78 
79 

852 

943 

68  034 

861 
952 
043 

870 
961 
052 

879 
970 
061 

888 
979 
070 

897 
988 
079 

906 

997 
088 

916 

*006 

097 

925 

*015 

10(5 

934 

*024 

115 

7 
8 
9 

7.0 
8.0 
9.0 

6.3 

7.2 
8.1 

5.6 
6.4 

7.2 

480 

124 

133 

142 

151 

1(30 

169 

178 

187 

196 

205 

81 

82 
83 

215 
305 
395 

224 
314 
404 

233 
323 
413 

242 
332 
422 

251 
341 
431 

260 
350 
440 

269 
359 
449 

278 
368 
458 

287 
377 
467 

296 
386 
476 

84 

85 
86 

485 
574 
664 

494 

583 
673 

502 
592 
681 

nil 

601 
690 

520 
610 
699 

529 
619 

708 

538 
628 
717 

547 
637 
726 

556 
646 
735 

565 
655 
744 

87 
88 
89 

753 

842 
931 

762 
851 
940 

771 

860 
949 

780 
869 
958 

789 
878 
966 

797 
886 
975 

806 
895 
984 

815 
904 
993 

824 

913 

*002 

833 

922 

*011 

490 

69  020 

028 

037 

046 

055 

064 

073 

082 

090 

099 

91 
92 
93 

108 
197 

285 

117 

205 
294 

126 
214 
302 

135 
223 
311 

144 
232 
320 

152 
241 

329 

161 
249 
338 

170 

258 
346 

179 
267 
355 

188 
276 
364 

94 
95 
96 

373 
461 
548 

381 
469 
557 

390 
478 
566 

399 
487 
574 

408 
496 
583 

417 
504 
592 

425 
513 
601 

434 
522 
609 

443 
531 

618 

452 
539 
627 

97^' 
98 
99 

636 
723 

810 

644 

732 
819 

653 
740 

827 

662 

749 
836 

671 

758 
845 

679 
767 
854 

688 
775 

862 

697 

784 
871 

705 
793 

880 

714 

801 
888 

500 

897 

906 

914 

923 

932 

940 

949 

958 

966 

975 

N. 

'    0 

1 

2 

3 

4 

5 

6 

7  1 

8 

9 

Prop.  Pts. 

10 

500- 

-  Logarithms  of  Numbers 

-550 

[I 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

500 

69  897 

906 

914 

923 

932 

940 

949 

958 

966 

975 

log  5=  .6989700043 

01 
02 
03 

984 

70  070 

157 

992 
079 
165 

*001 
088 
174 

*010 
096 
183 

*018 
105 
191 

*027 
114 
200 

*036 
122 
209 

*044 
131 
217 

*053 
140 
226 

*062 
148 
234 

04 
05 
06 

243 
329 
415 

252 

338 
424 

260 
346 
432 

269 
355 
441 

278 
364 
449 

286 
372 
458 

295 
381 
467 

303 
389 
475 

312 

398 
484 

321 
406 
492 

07 
08 
09 

501 
586 
672 

509 
595 
680 

518 
689 

52(7 
612 
697 

535 
621 
706 

544 
629 
714 

552 
638 
723 

661 
646 
731 

.569 
655 
740 

578 
663 
749 

510 

757 

766 

774 

783 

791 

800 

808 

817 

825 

834 

11 
12 
13 

842 

927 

71012 

851 
935 
020 

859 
944 
029 

868 
952 
037 

876 
961 
046 

885 
969 
054 

893 
978 
063 

902 
986 
071 

910 
995 
079 

919 

*003 
088 

14 
15 
16 

096 
181 
265 

105 
189 
273 

113 
198 

282 

122 

206 
290 

130 
214 
299 

139 
223 

307 

147 
231 
315 

155 
240 
324 

164 

248 
332 

172 
257 
341 

17 
18 
19 

349 
433 
517 

357 
441 
525 

366 
450 
533 

374 

458 
542 

383 
466 
550 

391 
475 

559 

399 
483 
567 

408 
492 
575 

416 
500 
584 

425 

508 
592 

520 

600 

609 

617 

625 

634 

642 

650 

659 

667 

675 

21 
22 

23 

684 
767 
850 

692 
775 

858 

700 
784 
867 

709 
792 

875 

717 
800 
883 

725 

809 
892 

734 
817 
900 

742 
825 
908 

750 
834 
917 

759 
842 
925 

1 

2 

9 

0.9 
1.8 

8 

0.8 
1.6 

7 

0.7 
1.4 

24 
25 
26 

933 

72016 

099 

941 
024 

107 

950 
032 
115 

958 
041 
123 

966 
049 
132 

975 
057 
140 

983 
066 
148 

991 
074 
156 

999 
082 
165 

*008 
O^X) 
173 

3 
4 
5 
6 

2.7 
3.6 
4.5 
5  4 

2.4 
3.2 
4.0 

4  8 

2.1 
2.8 
3.5 
4.2 
4.9 
5.6 
6.3 

27 
28 
29 

181 
263 
346 

189 

272 
354 

198 
280 
362 

206 
288 
370 

214 

296 

378 

222 

304 

387 

230 
313 
395 

239 
321 
403 

247 
329 
411 

255 
337 
419 

7 
8 
9 

6.3 

7.2 
8.1 

5.6 
6.4 

7.2 

530 

428 

436 

444 

452 

460 

469 

477 

485 

493 

501 

31 
32 
33 

509 
591 
673 

518 
599 
681 

526 
607 
689 

534 
616 
697 

542 
624 
705 

550 
.'532 
713 

558 
640 
722 

567 
648 
730 

575 
656 

738 

683 
665 
746 

34 
35 
36 

754 
835 
916 

762 
843 
925 

770 
852 
933 

779 
860 
941 

787 
868 
949 

795 
876 
957 

803 
884 
965 

811 
892 
973 

819 
900 
981 

827 
908 
989 

37 
38 
39 

997 

73078 

159 

*006 
086 
167 

*014 
094 
175 

*022 
102 
183 

*030 
111 
191 

*038 
119 
199 

*046 
127 
207 

*054 
135 
215 

*062 
143 
223 

*070 
151 
231 

540 

239 

247 

255 

263 

272 

280 

288 

296 

304 

312 

41 
42 
43 

320 
400 
480 

328 
408 
488 

336 
416 
496 

344 
424 

504 

352 
432 
512 

360 
440 
520 

368 
448 

528 

376 
456 
536 

384 
464 
544 

392 

472 
552 

44 
45 
46 

560 
640 
719 

568 
648 

727 

576 
656 
735 

584 
664 
743 

592 
672 
751 

600 
679 
759 

608 
687 
767 

616 
695 

775 

624 
703 
783 

632 
711 
791 

47 
48 
49 

799 
878 
957 

807 
886 
965 

815 
894 
973 

823 
902 
981 

830 
910 
989 

838 
918 
997 

846 

926 

*005 

854 

933 

*013 

862 

941 

*020 

870 
949 

*028 

650 

74036 

044 

052 

060 

068 

076 

084 

092 

099 

107 

N. 

0   1  1 

2  1  3 

4 

5 

6 

7 

8 

9 

Prop.  Pts* 

q 

550- 

-Logarithms  of  Numbers 

—  600 

11 

N= 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

560 

74  036 

044 

052 

060 

068 

076 

084 

092 

099 

107 

51 
52 
53 

115 
194 
273 

123 
202 

280 

131 
210 

288 

139 

218 
296 

147 
225 
304 

155 
233 
312 

162 
241 
320 

170 
249 
327 

178 
257 
335 

186 
265 
343 

54 
55 
56 

351 
429 
507 

359 
437 
515 

367 
445 
523 

374 
453 
531 

382 
461 
539 

390 
468 
547 

398 
476 
554 

406 

484 
562 

414 
492 
570 

421 
500 
578 

57 
58 
59 

586 
663 
741 

593 

671 
749 

601 
679 

757 

609 
687 
764 

617 
695 

772 

624 
702 

780 

632 
710 

788 

640 
718 
796 

648 
726 
803 

656 
733 
811 

560 

819 

827 

834 

842 

850 

858 

865 

873 

881 

889 

61 
62 
63 

896 

974 

75  051 

904 
981 
059 

912 

989 
066 

920 
997 
074 

927 
*005 

082 

935 

*012 
089 

943 

*020 

097 

950 

*028 

105 

958 

*035 

113 

966 

*043 

120 

64 
65 

m 

128 
205 

282 

136 
213 

289 

143 

220 
297 

151 

228 
305 

159 
236 
312 

166 
243 
320 

174 
251 

328 

182 
259 
335 

189 
266 
343 

197 
274 
351 

67 
68 
69 

358 
435 
511 

366 
442 
519 

374 
450 

526 

381 
458 
534 

389 
465 
542 

397 
,473 
549 

404 
481 
557 

412 
488 
565 

420 
496 
572 

427 
504 

580 

570 

587 

595 

603 

610 

618 

62() 

633 

641 

648 

656 

71 

72 
73 

6()4 
740 
815 

671 

747 
823 

679 
755 
831 

686 

762 
838 

694 
770 
846 

702 
778 
853 

709 

785 
861 

717 
793 
868 

724 

800 
876 

732 

808 
884 

1 

2 

8 

0.8 
1.6 

7 

0.7 
1.4 

74 
75 
76 

891 

967 

76042 

899 
974 
050 

906 
982 
057 

914 
989 
065 

921 

997 
072 

929 

*005 

080 

937 

*012 

087 

944 

*020 
095 

952 

*027 

103 

959 

*035 

110 

3 
4 
5 
6 

2.4 
3.2 
4.0 

4.8 

2.1 

2.8 
3.6 
4.2 

77 
78 
79 

580 

118 
193 

268 

125 
200 
275 

133 

208 
283 

140 
215 

290 

148 
223 

298 

155 

2;30 
305 

163 

238 
313 

170 

245 
320 

178 
253 
328 

185 
260 
335 

7 
8 
9 

6.6 
6.4 

7.2 

4.9 
5.6 
6.3 

343 

350 

358 

365 

373 

380 

388 

395 

403 

410 

81 

82 
83 

418 
492 
567 

425 
500 
574 

433 
507 

682 

440 
515 
589 

448 
522 
597 

455 
530 
604 

462 
537 
612 

470 
545 
619 

477 
552 
626 

485 
559 
634 

84 
85 
86 

641 

716 
790 

649 
723 

797 

656 
730 
805 

664 
738 
812 

671 

745 
819 

678 
753 
827 

686 
760 
834 

693 
768 
842 

701 

775 
849 

708 
782 
856 

87 
88 
89 

864 

938 

77  012 

871 
945 
019 

879 
953 
026 

886 
960 
034 

893 
967 
041 

901 
975 
048 

908 
982 
0:>6 

916 

989 
063 

923 

997 
070 

930 

*004 

078 

590 

085 

093 

100 

107 

115 

122 

129 

137 

144 

151 

91 
92 
93 

159 
232 
305 

166 
240 
313 

173 

247 
320 

181 
254 
327 

188 
262 
335 

195 

269 
342 

203 
276 
349 

210 

283 
357 

217 
291 
364 

225 

298 
371 

94 
95 
96 

379 
452 
525 

386 
459 
532 

393 
466 
539 

401 
474 
546 

408 

481 
554 

415 

488 
561 

422 
495 
568 

430 
503 
576 

437 

510 
583 

444 
517 
690 

97. 
98 
99 

597 
670 
743 

605 
677 
750 

612 

685 

757 

619 
692 
764 

627 
699 

772 

634 
706 
779 

641 
714 

786 

648 
721 

793 

656 

728 
801 

663 
735 

808 

600 

815 

822 

830 

837^ 

844 

851 

859 

866 

873 

880 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

12 

600- 

-Logarithms  of  Numbers 

—  650 

[1 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

600 

77  815 

822 

830 

837 

844 

851 

859 

866 

873 

880 

01 
02 
03 

887 

960 

78032 

895 
967 
039 

902 
974 
046 

909 
981 
053 

916 
988 
061 

924 

996 
068 

931 

*003 

075 

938 

*010 

082 

945 

*017 

089 

952 

*025 

097 

04 
05 
06 

104 
176 

247 

111 
183 
254 

118 
l^X) 
262 

125 
197 
269 

132 
204 
276 

140 
211 

283 

147 
219 
290 

154 
226 

297 

161 
233 

305 

168 
240 
312 

07 
08 
09 

319 
390 
462 

326 
398 
469 

333 
405 

476 

aio 

412 

483 

347 
419 
490 

355 
426 

497 

362 
433 
504 

369 
440 
512 

376 
447 
519 

383 
455 
526 

610 

533 

540 

547 

554 

561 

569 

576 

583 

590 

597 

11 
12 
13 

604 
675 
746 

611 

682 
753 

618 
689 
760 

625 
696 
767 

633 

704 

774 

640 
711 

781 

647 
718 

789 

654 

725 
796 

661 
732 
803 

668 
739 
810 

14 
15 
16 

817 
888 
958 

824 
895 
965 

831 
902 
972 

838 
909 
979 

845 
916 
986 

852 
923 
993 

859 

930 

*000 

866 

937 

*007 

873 

944 

*014 

880 

951 

*021 

17 

18 
19 

79029 
099 
169 

036 
106 
176 

043 
113 
183 

050 
120 
190 

057 
127 
197 

064 
134 
204 

071 
141 

211 

078 

•148 

218 

085 
155 
225 

092 
162 
232 

620 

239 

246 

253 

260 

267 

274 

281 

288 

295 

302 

21 
22 
23 

309 
379 
449 

316 
386 
456 

323 
393 
463 

330 
400 
470 

337 
407 

477 

344 
414 

484 

351 
421 
491 

358 
428 
498 

365 
435 
505 

372 
442 
511 

1 

2 

8 

0.8 
1.6 

7 

0.7 
1.4 

6 

0.6 
1.2 

24 
25 
26 

518 
588 
657 

525 

595 
664 

532 
602 
671 

539 
609 
678 

546 
616 
685 

553 
623 
692 

560 
630 
699 

567 
637 
706 

574 
644 
713 

581 
650 
720 

3 
4 
5 
6 

2.4 
3.2 
4.0 

48 

2.1 

2.8 
3.5 

4  2 

1.8 
2.4 
3.0 
3.6 

27 
28 
29 

727 
796 
865 

734 

803 

872 

741 
810 

879 

748 
817 
886 

754 
824 
893 

761 
831 

900 

768 
837 
906 

775 
844 
913 

782 
851 
920 

789 
858 
927 

7 
8 
9 

5.6 
6.4 
7.2 

4.9 
5.6 
6.3 

4.2 
4.8 
5.4 

630 

934 

941 

948 

955 

962 

969 

975 

982 

989 

996 

31 
32 
33 

80003 
072 
140 

010 
079 
147 

017 
085 
154 

024 
092 
161 

030 
099 

168 

037 
106 
175 

044 
113 
182 

051 
120 

188 

058 
127 
195 

065 
134 
202 

34 
35 
36 

209 

277 
346 

216 
284 
353 

223 
291 
359 

229 

298 
366 

236 
305 
373 

243 
312 

380 

250 

318 
387 

257 
325 
393 

264 
532 
400 

271 

339 
407 

37 
38 
39 

414 

482 
550 

421 
489 
557 

428 
496 
564 

434 
502 
570 

441 
509 

577 

448 
516 

584 

455 
523 
591 

462 
530 

598 

468 
536 
604 

475 
543 
611 

640 

618 

625 

632 

638 

645 

652 

659 

665 

672 

679 

41 
42 
43 

686 
754 
821 

693 
760 
828 

699 
767 
835 

706 

774 
841 

713 

781 
848 

720 

787 
855 

726 
794 
862 

733 

801 
868 

740 
808 
875 

747 
814 
882 

44 
45 
46 

889 

956 

81023 

895 
963 
030 

902 
969 
037 

909 
976 
043 

916 
983 
050 

922 

990 
057 

929 

996 
064 

936 

*003 

070 

943 
*010 

077 

949 
*017 
084 

47 

48 
49 

090 
158 
224 

097 
164 
231 

104 
171 

238 

111 
178 
245 

117 
184 
251 

124 
191 

258 

131 
198 
265 

137 
204 
271 

144 
211 

278 

151 
218 

285 

650 

291 

298 

305 

311 

318 

325 

331 

338 

345 

351 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

i] 

650- 

-Logarithms  of  Numbers 

—  700 

13 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pt6. 

650 

81291 

298 

305 

311 

318 

325 

331 

338 

345 

351 

51 

358 

365 

371 

378 

385 

391 

398 

405 

411 

418 

52 

425 

431 

438 

445 

451 

458 

465 

471 

478 

485 

53 

491 

498 

505 

511 

518 

625 

631 

538 

544 

651 

54 

558 

564 

571 

578 

584 

591 

598 

604 

611 

617 

55 

624 

631 

637 

644 

651 

657 

664 

671 

677 

684 

56 

690 

697 

704 

710 

717 

723 

730 

737 

743 

750 

57 

757 

763 

770 

776 

783 

790 

796 

803 

809 

816 

58 

823 

829 

836 

842 

849 

856 

862 

869 

875 

882 

59 
660 

889 

895 

902 

908 

915 

921 

928 

935 

941 

948 

954 

961 

968 

974 

981 

987 

994 

*000 

*007 

*014 

Gl 

82020 

027 

033 

040 

046 

053 

060 

066 

073 

079 

()2 

086 

092 

099 

105 

112 

119 

125 

132 

138 

145 

63 

151 

158 

164 

171 

178 

184 

191 

197 

204 

210 

64 

217 

223 

230 

236 

243 

249 

256 

263 

269 

276 

65 

282 

289 

295 

302 

308 

315 

321 

328 

334 

341 

m 

347 

354 

360 

367 

373 

380 

387 

393 

400 

406 

67 

413 

419 

426 

432 

439 

445 

452 

458 

465 

471 

68 

478 

484 

491 

497 

504 

510 

517 

623 

530 

536 

69 

643 

549 

556 

562 

569 

575 

582 

688 

595 

601 

670 

607 

614 

620 

627 

633 

640 

646 

653 

659 

666 

71 

672 

679 

685 

692 

698 

705 

711 

718 

724 

730 

7 

6 

72 

737 

743 

750 

756 

763 

769 

776 

782 

789 

795 

1 

0.7 

0.6 

73 

802 

808 

814 

821 

827 

834 

840 

847 

853. 

860 

2 

1.4 

1.2 

74 

866 

872 

879 

885 

892 

898 

905 

911 

918 

924 

3 
4 
5 
6 

2.1 
2.8 
3.5 
4.2 

1.8 
2.4 
3.0 
3.6 

75 

930 

937 

943 

950 

956 

963 

969 

975 

982 

988 

76 

995 

*001 

*008 

*014 

*020 

*027 

*033 

*040 

*046 

*052 

77 

83059 

065 

072 

078 

085 

091 

097 

104 

110 

117 

7 

4.9 

4.2 

78 

123 

129 

136 

142 

149 

155 

161 

168 

174 

181 

8 

5.6 

4.8 

79 

187 

193 

200 

206 

213 

219 

225 

232 

238 

245 

9 

6.3 

5.4 

680 

81 

251 

257 

264 

270 

276 

283 

289 

296 

302 

308 

315 

321 

327 

334 

340 

347 

353 

359 

366 

372 

82 

378 

385 

391 

398 

404 

410 

417 

423 

429 

436 

83 

442 

448 

455 

461 

467 

474 

480 

487 

493 

499 

84 

506 

512 

518 

525 

531 

537 

544 

550 

556 

563 

85 

569 

575 

582 

588 

594 

601 

607 

613 

620 

626 

86 

632 

639 

645 

651 

658 

664 

670 

677 

683 

689 

87 

696 

702 

708 

715 

721 

727 

734 

740 

746 

753 

88 

759 

765 

771 

778 

784 

790 

797 

803 

809 

816 

89 

822 

828 

835 

841 

847 

853 

860 

866 

872 

879 

690 

885 

891 

897 

904 

910 

916 

923 

929 

935 

942 

91 

948 

954 

960 

967 

973 

979 

985 

992 

998 

*004 

92 

84  011 

017 

023 

029 

036 

042 

048 

055 

061 

067 

93 

073 

080 

086 

092 

098 

105 

111 

117 

123 

130 

94 

136 

142 

148 

155 

161 

167 

173 

180 

186 

192 

95 

198 

205 

211 

217 

223 

230 

236 

242 

248 

255 

96 

261 

267 

273 

280 

286 

292 

298 

305 

311 

317 

97 

323 

330 

336 

342 

348 

354 

361 

367 

373 

379 

98 

386 

392 

398 

404 

410 

417 

423 

429 

435 

442 

99 

448 

454 

460 

466 

473 

479 

485 

491 

497 

504 

700 

510 

516 

522 

528 

535 

541 

547 

553 

559 

566 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

14 

7 

00- 

-  Logarithms  of  Numbers 

-760 

[I 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

700 

84  510 

516 

522 

528 

535 

541 

547 

553 

559 

566 

log  7-  .8450980400 

01 
02 
03 

572 
634 
696 

578 
640 
702 

584 
646 
708 

590 
652 
714 

597 
658 
720 

603 
665 

726 

609 
671 
733 

615 

677 
739 

621 
683 
745 

628 
689 
751 

04 
05 
06 

757 
819 
880 

763 
825 
887 

770 
831 
893 

776 
837 
899 

782 
844 
905 

788 
850 
911 

794 
856 
917 

800 
862 
924 

807 
868 
930 

813 
874 
936 

07 
08 
09 

942 

85003 
065 

948 
009 
071 

954 
016 
077 

mo 

022 

083 

967 
028 
089 

973 

034 
095 

979 
040 
101 

985 
046 
107 

991 
052 
114 

997 
058 
120 

710 

126 

132 

138 

144 

150 

156 

163 

169 

175 

181 

11 
12 
13 

187 
248 
309 

193 
254 
315 

199 
260 
321 

205 
266 
327 

211 
272 
333 

217 
278 
339 

224 

285 
345 

230 
291 
352 

236 
297 
358 

242 

303 
364 

14 
15 
16 

370 
431 
491 

376 
437 
497 

382 
443 
503 

388 
449 
509 

394 
455 
516 

400 
461 
522 

406 
467 
528 

412 
473 
534 

418 
479 
540 

425 

485 
546 

17 
18 
19 

720 

552 
612 
673 

558 
618 
679 

564 

()25 
685 

570 
631 
691 

576 
637 
697 

582 
643 
703 

588 
649 
709 

594 
655 
715 

600 
661 
721 

606 
667 

727 

733 

739 

745 

751 

757 

763 

769 

775 

781 

788 

21 
22 
23 

794 
854 
914 

800 
860 
920 

806 
866 
926 

812 
872 
962 

818 
878 
938 

824 
884 
944 

830 
890 
950 

836 
896 
956 

842 
^)02 
962 

848 
908 
968 

1 

2 

7 

0.7 
1.4 

6 

0.6 
1.2 

5 

0.5 
1.0 

24 
25 
26 

974 

86034 

094 

980 
040 
100 

986 
046 
106 

902 
052 
112 

998 
058 
118 

*004 
064 
124 

*010 
070 
130 

*016 
076 
136 

*022 
082 
141 

*028 
088 
147 

3 
4 
5 
6 

2.1 

2.8 
3.5 
4*^ 

1.8 
2.4 
3.0 
36 

1.5 
2.0 
2.5 
30 

27 
28 
29 

730 

153 
213 

273 

332 

159 
219 

2<9 

165 
225 

285 

171 

231 
291 

177 

237 

297 

183 
243 
303 

189 
249 

308 

195 
255 
314 

201 
261 
320 

207 

267 
326 

7 
8 
9 

4.9 
5.6 
6.3 

4.2 

4.8 
5.4 

3.5 
4.0 
4.5 

338 

344 

350 

356 

362 

368 

374 

380 

386 

31 
32 
33 

392 
451 
510 

398 
457 
516 

404 
463 
522 

410 

4()9 
528 

415 
475 
534 

421 
481 
540 

427 
487 
546 

433 
493 
552 

439 
499 
558 

445 
604 
564 

34 
35 
36 

570 
629 

688 

576 
635 
694 

581 
641 
700 

587 
646 
705 

593 
652 
711 

599 
658 

717 

605 
6t>4 
723 

611 
670 
729 

617 
676 
735 

623 
682 
741 

37 
38 
39 

747 
806 
864 

753 

812 
870 

759 

817 
876 

764 
823 

882 

770 

829 

888 

776 

835 
894 

782 
841 
fX)0 

788 
847 
906 

794 
853 
911 

800 
859 
917 

740 

923 

929 

935 

941 

947 

953 

958 

964 

970 

976 

41 
42 
43 

982 

87040 

099 

988 
046 
105 

994 
052 
111 

999 
058 
116 

*005 
064 
122 

*011 
070 
128 

*017 
075 
134 

*023 
081 
140 

*029 
087 
146 

*035 
093 
151 

44 
45 
46 

157 
216 
274 

163 
221 

280 

169 
227 
286 

175 
233 
291 

181 
239 
297 

186 
245 
303 

192 
251 
309 

198 
256 
315 

204 
262 
320 

210 
268 
326 

47 

48 
49 

332 
390 
448 

338 
396 
454 

344 
402 
460 

349 

408 
466 

355 
413 

471 

361 
419 
477 

367 
425 

483 

373 
431 

489 

379 

437 
495 

552 

384 
442 
500 

750 

506 

512 

518 

523 

529 

535 

.541 

547 

558 

N. 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

I] 

760- 

-  Logarithms  of  Numbers 

—  800 

15 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

750 

87  506 

512 

518 

523 

529 

535 

541 

547 

552 

558 

51 
52 
53 

564 
622 
679 

570 

628 
685 

576 
633 
691 

581 
639 
697 

587 
645 
703 

593 
651 
708 

599 
656 
714 

604 
()62 
720 

610 
6f)8 
726 

616 
674 
731 

54 
55 
56 

737 
795 
852 

743 

800 
858 

749 
806 
864 

754 
812 

869 

760 

818 

875 

766 
823 

881 

772 
829 
887 

777 
835 
892 

783 
841 
898 

789 
846 
904 

57 
58 
59 

910 

967 

88  024 

915 
973 
030 

921 

978 
036 

927 
984 
041 

933 

990 
047 

938 
996 
053 

944 

*001 

058 

950 

*007 

064 

955 

*013 

070 

961 

*018 

076 

760 

081 

087 

093 

098 

104 

110 

116 

121 

127 

133 

61 

62 
63 

138 
195 
252 

144 
201 
258 

150 
207 
264 

156 
213 
270 

161 
218 
275 

167 
224 

281 

173 
230 

287 

178 
235 
292 

184 
241 

298 

190 
247 
304 

64 
65 

m 

309 
366 
423 

315 
372 
429 

321 
377 
434 

326 
383 
440 

332 
389 
446 

338 
395 
451 

343 

400 
457 

349 
406 
463 

355 
412 
468 

360 
417 
474 

67 
68 
69 

480 
536 
593 

485 
542 
598 

491 
547 
604 

497 
553 
610 

502 
559 
615 

508 
564 
621 

513 

570 
()27 

519 
576 
632 

525 

581 

638 

530 
587 
643 

770 

649 

655 

660 

666 

672 

677 

683 

689 

694 

700 

71 
72 
73 

705 
762 
818 

711 

767 
824 

717 
773 
829 

722 
779 
835 

728 
784 
840 

734 
190 
846 

739 
795 
852 

745 
801 
857 

750 
807 
863 

756 
812 
868 

1 

2 

6 

0.6 
1.2 

5 

0.5 
1.0 

74 
75 
76 

874 
930 

986 

880 
936 
992 

885 
941 
997 

891 

947 

*003 

897 

953 

*009 

902 

958 

*014 

908 

9CA 

*020 

913 

*025 

919 

975 
*031 

925 

981 

*037 

3 
4 
5 
6 

1.8 
2.4 
3.0 
3.6 

1.5 
2.0 
2.5 
3  0 

77 
78 
79 

89042 
098 
154 

048 
104 
159 

053 
109 
165 

059 
115 
170 

064 
120 

176 

070 
126 
182 

076 
131 

187 

081 
137 
193 

087 
143 
198 

254 

092 
148 
204 

7 
8 
9 

4.2 
4.8 
5.4 

3.5 
4.0 
4.5 

780 

209 

215 

221 

226 

232 

237 

243 

248 

260 

81 
82 
83 

265 
321 
376 

271 
326 

382 

276 
332 
387 

282 
337 
393 

287 
343 
398 

293 
348 
4(H 

298 
354 
409 

304 
360 
415 

310 
365 
421 

315 
371 

426 

84 

85 
86 

432 
487 
542 

437 
492 

548 

443 
498 
653 

448 
504 
559 

454 
509 
564 

459 
515 
570 

465 
520 
575 

470 
526 
581 

476 

531 

586 

481 
537 
592 

87 
88 
89 

597 
653 

708 

603 
658 
713 

609 
664 
719 

614 
669 
724 

620 

675 
730 

625 
680 
735 

631 

686 
741 

636 
691 
746 

642 

697 

752 

647 
702 
757 

790 

763 

768 

774 

779 

785 

790 

796 

801 

807 

812 

91 
92 
93 

818 
873 
927 

823 
878 
933 

829 
883 
938 

834 
889 
944 

840 
894 
949 

845 
900 
955 

851 
905 
960 

856 
911 
966 

862 
916 
971 

867 
922 
977 

94 
95 
96 

982 

90037 

091 

988 
042 
097 

993 
048 
102 

998 
053 
108 

*004 
059 
113 

*009 
064 
119 

*015 
069 
124 

*020 
075 
129 

*026 
080 
135 

*031 
086 
140 

97 
98 
99 

146 
200 
255 

151 
206 
260 

157 
211 

266 

162 
217 
271 

168 
222 
276 

173 

227 

282 

179 
233 

287 

184 
238 
293 

189 
244 

298 

195 

249 
304 

800 

309 

314 

320 

325 

331 

336 

342 

347 

352 

358 

U. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

16 

800- 

-  Logarithms  of  Numbers 

-850 

[1 

N, 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

800 

90  309 

314 

320 

325 

331 

336 

342 

347 

352 

358 

01 
02 
03 

363 

417 

472 

369 
423 

477 

374 
428 
482 

380 
434 
488 

385 
439 
493 

390 
445 
499 

396 
450 
504 

401 
455 
509 

407 
461 
515 

412 
466 
620 

04 
05 
06 

526 

580 
634 

531 

585 
639 

536 
590 
644 

542 
596 
650 

547 
601 
655 

553 
607 
660 

558 
612 
666 

563 
617 
671 

569 
623 
677 

574 

628 
682 

07 
08 
09 

687 
741 
795 

693 

747 
800 

698 
752 
806 

703 

757 
811 

709 
763 
816 

714 

768 
822 

720 
773 

827 

725 
779 
832 

730 

784 
838 

736 

789 
843 

810 

849 

854 

859 

865 

870 

875 

881 

886 

891 

897 

11 
12 
13 

902 

956 

91009 

907 
961 
014 

913 
966 
020 

918 
972 
025 

924 

977 
030 

929 

982 
036 

934 
988 
041 

940 
993 
046 

945 
998 
052 

950 

*004 

057 

14 
15 
16 

062 
116 
169 

068 
121 
174 

073 
126 
180 

078 
132 
185 

084 
137 
190 

089 
142 
196 

094 
148 
201 

100 
153 
206 

105 
158 
212 

110 
164 
217 

17 
18 
19 

222 
275 

328 

228 
281 
334 

233 

286 
339 

238 
291 
344 

243 

297 
350 

249 
302 
355 

254 
307 
360 

259 
312 
365 

265 
318 
371 

270 
323 
376 

820 

381 

387 

392 

397 

403 

408 

413 

418 

424 

429 

21 

22 
23 

434 

487 
540 

440 
492 
545 

445 
498 
551 

450 
503 
656 

455 
508 
561 

461 
514 
566 

466 
519 
572 

471 
524 

577 

477 
529 
582 

482 
535 

587 

1 

2 

6 

0.6 
1.2 

5 

0.5 
1.0 

24 
25 
26 

593 
645 

698 

598 
651 
703 

603 
656 
709 

609 
661 
714 

614 
666 
719 

619 
672 

724 

624 

677 
730 

630 
682 
735 

635 

687 
740 

640 
693 
745 

3 
4 
5 
6 

1.8 
2.4 
3.0 
3.6 

1.5 
2.0 

2.5 
3.0 

27 
28 
29 

751 
803 
855 

756 
808 
861 

761 
814 
866 

766 
819 
871 

772 
824 
876 

777 
829 
882 

782 
834 
887 

787 
840 
892 

793 
845 
897 

798 
850 
903 

7 
8 
9 

4.2 
4.8 
6.4 

3.5 
4.0 
4.5 

830 

908 

913 

918 

924 

929 

934 

939 

944 

950 

955 

31 
32 
33 

960 

92012 

065 

965 
018 
070 

971 
023 
075 

976 

028 
080 

981 
033 

085 

986 
038 
091 

991 
044 
096 

997 
049 
101 

*002 
054 
106 

*007 
059 
111 

34 
35 
36 

117 
169 
221 

122 
174 
226 

127 
179 
231 

132 

184 
236 

137 
189 
241 

143 
195 

247 

148 
200 
252 

153 

205 
257 

158 
210 
262 

163 
215 

267 

37 
38 
39 

273 
324 
376 

278 
330 
381 

283 
335 

387 

288 
340 
392 

293 
345 

397 

298 
350 
402 

304 
355 
407 

309 
361 
412 

314 

366 
418 

319 
371 
423 

840 

428 

433 

438 

443 

449 

454 

459 

464 

469 

474 

41 
42 
43 

480 
531 
583 

485 
536 

588 

490 
542 
593 

495 
547 
598 

500 
552 
603 

505 
557 
609 

511 

562 
614 

516 
567 
619 

521 
572 
624 

526 

578 
629 

44 
45 
46 

634 

686 
737 

639 
691 
742 

645 
696 

747 

650 
701 
752 

655 
706 

758 

660 
711 
763 

665 
716 
768 

670 
722 
773 

675 

727 
778 

681 
732 
783 

47 
48 
49 

788 
840 
891 

793 
845 
896 

799 
850 
901 

804 
855 
906 

809 
860 
911 

814. 
865 
916 

819 
870 
921 

824 

875 
927 

829 
881 
932 

834 
886 
937 

850 

942 

947 

952 

957 

962 

967 

973 

978 

983 

988 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts.     . 

q 

8 

50- 

-Lo^ 

?arit 

hms 

of  Numbers 

—  900 

17 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

850 

92  942 

947 

952 

957 

962 

967 

973 

978 

983 

988 

51 
52 
53 

993 

93  044 

095 

998 
049 
100 

*003 
054 
105 

*008 
059 
110 

*013 
064 
115 

*018 
069 
120 

*024 
075 
125 

*029 
080 
13J 

*034 
085 
136 

*039 
090 
141 

54 
55 
56 

146 
197 
247 

151 
202 
252 

156 
207 
258 

161 

212 
263 

166 
217 
268 

171 
222 
273 

176 

227 
278 

181 
232 

283 

186 
237 
288 

192 
242 
293 

57 
58 
59 

298 
349 
399 

303 
354 
404 

308 
359 
409 

313 

364 
414 

318 
369 
420 

323 
374 
425 

328 
379 
430 

334 
384 
435 

339 
389 
440 

344 
394 
445 

860 

450 

455 

460 

465 

470 

475 

480 

485 

490 

495 

61 
62 
63 

500 
551 
601 

505 

556 
606 

510 
561 
611 

515 
560 
616 

520 
571 
621 

526 
576 
626 

531 
581 
631 

536 
586 
636 

541 
591 
641 

546 
596 
646 

64 
65 
66 

651 
702 
752 

656 
707 
757 

661 
712 

762 

666 
717 
767 

671 

722 
772 

676 

727 
777 

682 
732 

782 

687 
737 

787 

692 

742 
792 

697 

747 
797 

67 
68 
69 

802 
852 
902 

807 
857 
907 

812 
862 
912 

817 
867 
917 

822 

872 
922 

827 
877 
927 

832 
882 
932 

837 
887 
937 

842 
892 
942 

847 
897 
947 

997 

870 

952 

957 

962 

967 

972 

977 

982 

987 

992 

71 
72 
73 

94  002 
052 
101 

007 
057 
106 

012 
062 
111 

017 
067 
116 

022 
072 
121 

027 

077 
126 

032 

082 
131 

037 
086 
136 

042 
091 
141 

047 
096 
146 

1 
2 

6 

0.6 
1.2 

5 

0.5 
1.0 

4 

0.4 
0.8 

74 
75 
76 

151 
201 
250 

156 
206 
255 

161 
211 

260 

166 
216 
265 

171 
221 
270 

176 
226 
275 

181 
231 

280 

186 
236 

285 

191 

240 
290 

196 
245 
295 

3 
4 
5 

1.8 
2.4 
3.0 
3  6 

1.5 
2.0 
2.5 
30 

1.2 
1.6 
2.0 
2.4 

77 
78 
79 

300 
349 
399 

305 
354 
404 

310 
359 
409 

315 
364 
414 

320 
369 
419 

325 
374 
424 

330 
379 
429 

335 
384 
433 

340 

389 
438 

345 
394 
443 

7 
8 
9 

4.2 
4.8 
5.4 

3.5 
4.0 
4.5 

2.8 
3.2 
3.6 

880 

448 

453 

458 

463 

468 

473 

478 

483 

488 

493 

81 
82 
83 

498 
547 
596 

503 
552 
601 

507 
557 
606 

512 
562 
611 

517 
567 
616 

522 
571 
621 

527 
576 
626 

532 
581 
630 

537 
586 
635 

542 
591 
640 

84 
85 
86 

645 
694 
743 

650 
699 
748 

655 
704 
753 

660 
709 
758 

665 
714 
763 

670 
719 
768 

675 
724 
773 

680 
729 
778 

685 
734 

783 

689 
738 
787 

87 
88 
89 

792 
841 

890 

797 
846 
895 

802 
851 
900 

807 
856 
f)05 

812 
861 
910 

817 
866 
915 

822 
871 
919 

827 
876 
924 

832. 
880 
929 

836 
885 
934 

890 

939 

944 

949 

954 

959 

ms 

968 

973 

978 

983 

91 
92 
93 

988 

95  036 

085 

993 
041 
090 

998 
046 
095 

*002 
051 
100 

*007 
056 
105 

*012 
061 
109 

*017 
066 
114 

*022 
071 
119 

*027 
075 
124 

*032 
080 
129 

94 
95 
96 

134 

182 
231 

139 

187 
236 

143 
192 
240 

148 
197 
245 

153 
202 
250 

158 
207 
255 

163 
211 
260 

168 
216 
265 

173 
221 

270 

177 
226 
274 

97 
98 
99 

279 
328 
376 

284 
332 
381 

289 
337 
386 

294 
342 
390 

299 
347 
395 

303 
352 
400 

308 
357 
405 

313 
361 
410 

318 
366 
415 

323 
371 
419 

900 

424 

429 

434 

439 

444 

448 

453 

458 

463 

468 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

18 

900- 

-  Logarithms  of  Numbers 

-950 

U 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop,  Pts. 

900 

95  424 

429 

434 

439 

444 

448 

453 

458 

463 

468 

01 
02 
03 

472 
521 
569 

477 
525 
574 

482 
530 
578 

487 
535 
583 

492 
540 
588 

497 
545 
593 

501 
550 
598 

506 
554 
602 

511 

559 
607 

516 
564 
612 

04 
05 
06 

617 
665 
713 

622 
670 
718 

626 
674 

722 

631 
679 

727 

636 
684 

732 

641 

689 
737 

646 
694 

742 

650 
698 
746 

655 
703 
751 

660 
708 
756 

07 
08 
09 

761 

809 
85(i 

766 
813 
861 

770 

818 
866 

775 
823 
871 

780 
828 
875 

785 
832 
880 

789 
837 
885 

794 
842 
890 

799 
847 
895 

804 
852 
899 

910 

904 

909 

914 

918 

923 

928 

933 

938 

942 

947 

11 
12 
13 

952 

999 

96  047 

957 

*004 

052 

961 

*009 

057 

966 

*014 

061 

971 

*019 
066 

976 

*023 

071 

980 

*028 

076 

985 

*033 

080 

990 

*038 

085 

995 

*042 

090 

14 
15 
16 

095 
142 
190 

099 
147 
194 

104 

152 
199 

109 
156 
204 

114 
161 
209 

118 
166 
213 

123 
171 

218 

128 
175 
223 

133 
180 
227 

137 
185 
232 

17 
18 
19 

237 

284 
332 

242 
289 
336 

246 
294 
341 

251 
298 
346 

256 
303 
350 

261 
308 
355 

265 
313 
360 

270 

317 

365 

275 
322 
369 

280 
327 
374 

920 

379 

384 

388 

393 

398 

402 

407 

412 

417 

421 

21 
22 
23 

426 
473 
520 

431 
478 
625 

435 
483 
530 

440 
487 
534 

445 

492 
539 

450 
497 
544 

454 
501 
548 

459 
506 
553 

464 
511 
558 

468 
515 
562 

1 

2 

5 

0.5 
1.0 

4 

0.4 

0.8 

24 
25 
26 

567 
614 
661 

572 
619 
666 

577 
624 
670 

581 
628 
675 

586 
633 

680 

591 
638 
685 

595 
642 
689 

600 
647 
694 

605 
652 
699 

609 
656 
703 

3 
4 
5 
6 

1.5 
2.0 
2.5 
3.0 

1.2 
1.6 
2.0 
2.4 

27 
28 
29 

708 
755 
802 

713 

759 
806 

717 
764 
811 

722 
769 

816 

727 
774 
820 

731 
778 

825 

736 

783 
830 

741 

788 
834 

745 

792 
839 

760 

797 
844 

7 
8 
9 

3.5 
4.0 
4.5 

2.8 
3.2 
3.6 

930 

848 

853 

858 

862 

867 

872 

876 

881 

886 

890 

31 
32 
33 

895 
94- 
988 

900 
946 
993 

904 
951 
997 

909 

956 

*002 

914 

960 

*007 

918 

965 

*011 

923 

970 

■*016 

928 

974 

*021 

932 

979 

*025 

937 

984 
*030 

34 
35 
36 

97035 
081 

128 

039 
086 
132 

044 
090 
137 

049 
095 
142 

053 
100 
146 

058 
104 
151 

063 
109 
155 

067 
114 
160 

072 

118 
165 

077 
123 
169 

37 
38 
39 

174 
220 
267 

179 
225 
271 

183 
230 
276 

188 
234 

280 

192 

239 

285 

197 
243 

290 

202 
248 
294 

206 
253 
299 

211 

257 
304 

216 
262 

308 

940 

41 
42 
43 

313 

359 
405 
451 

317 

322 

327 

331 

336 

340 

345 

350 

354 

364 
410 
456 

368 
414 
460 

373 
419 
465 

377 
424 
470 

382 
428 
474 

387 
433 
479 

391 
437 
483 

396 
442 

488 

400 
447 
493 

44 

45 
46 

497 
543 
589 

502 
548 
594 

506 
552 

598 

511 
557 
603 

516 

562 
607 

520 
566 
612 

525 
571 

617 

529 
575 
621 

534 
580 
626 

539 
585 
630 

47 
48 
49 

635 
(>81 

727 

640 
685 
731 

644 
690 
736 

649 
695 
740 

653 
699 
745 

658 
704 
749 

663 
708 
754 

667 
713 
759 

672 
717 
763 

676 

722 
768 

950 

772 

777 

782 

786 

791 

795 

800 

804 

809 

813 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

i] 

950  — 

Lo^arith 

ms  of  Numbers  - 

-1000 

19 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

950 

97  772 

777 

782 

786 

791 

795 

800 

804 

809 

813 

51 
52 
53 

818 
864 
909 

823 
868 
914 

827 
873 
918 

832 
877 
923 

836 
882 
928 

841 
886 
932 

845 
891 
937 

850 
896 
941 

855 
900 
946 

859 
905 
950 

54 
55 
56 

955 

98  000 

046 

959 
005 
050 

964 
009 
055 

968 
014 
059 

973 
019 
064 

978 
023 
068 

982 
028 
073 

987 
032 
078 

991 
037 
082 

996 
041 
087 

57 
58 
59 

091 
137 
182 

096 
141 

186 

100 
146 
191 

105 
150 
195 

109 
155 
200 

114 

159 
204 

118 
164 
209 

123 

168 
214 

127 
173 

218 

132 
177 
223 

960 

227 

232 

236 

241 

245 

250 

254 

259 

263 

268 

61 
62 
63 

272 

318 
363 

277 
322 
367 

281 
327 
372 

286 
331 
376 

290 
336 
381 

295 
340 

385 

299 
345 
390 

304 
349 
394 

308 
354 
399 

313 

358 
403 

64 
65 

m 

408 
453 
498 

412 

457 
502 

417 
462 
507 

421 
466 
511 

426 
471 
516 

430 
475 
520 

435 

480 
525 

439 

484 
529 

444 

489 
534 

448 
493 
538 

67 
68 
69 

543 

588 
632 

547 
592 
637 

552 
597 
641 

556 
601 
646 

561 

605 
650 

565 
610 
655 

570 
614 
659 

574 
619 
664 

579 
623 

668 

583 
628 
673 

970 

677 

682 

686 

691 

695 

700 

704 

709 

713 

717 

71 
72 
73 

722 
767 
811 

726 
771 
816 

731 
776 
820 

735 
780 
825 

740 

784 
829 

744 
789 
834 

749 
793 

838 

753 

798 
843 

758 
802 
847 

762 
807 
851 

1 
2 

5 

0.5 
1.0 

4 

0.4 
0.8 

74 
75 
76 

856 
900 
945 

860 
905 
949 

865 
909 
954 

869 
914 
958 

874 
918 
963 

878 
923 
967 

883 
927 
972 

887 
932 
976 

892 
936 
981 

896 
941 
985 

3 
4 
5 
6 

1.5 
2.0 
2.5 
3  0 

1.2 
1.6 
2.0 
2.4 

77 
78 
79 

989 
99  034 

078 

994 
038 

083 

998 
043 

087 

*003 
047 
092 

*007 
052 
096 

*012 
056 
100 

*016 
061 
105 

*021 
065 
109 

*025 
069 
114 

*029 
074 
118 

7 
8 
9 

3.5 
4.0 
4.5 

2.8 
3.2 
3.6 

980 

123 

127 

131 

136 

140 

145 

149 

154 

158 

162 

81 
82 
83 

167 
211 
255 

171 
216 
260 

176 
220 
264 

180 
224 
269 

185 
229 
273 

189 
233 

277 

193 

238 
282 

198 

242 

286 

202 
247 
291 

207 
251 
295 

84 

85 
86 

300 
344 

388 

304 
348 
392 

308 
352 
396 

313 
357 
401 

317 
361 
405 

322 
'666 
410 

326 
370 
414 

330 
374 
419 

335 
379 
423 

339 

383 
427 

87 
88 
89 

432 
476 
520 

436 
480 
524 

441 

484 

528 

445 
489 
533 

449 

493 
537 

454 
498 

542 

458 
502 
546 

463 

506 
550 

467 
511 
555 

471 
515 
559 

990 

564 

568 

572 

577 

581 

585 

590 

594 

599 

603 

91 
92 
93 

607 
651 
695 

612 
656 
699 

616 
660 
704 

621 

664 
708 

625 
669 
712 

629 
673 
717 

634 
677 
721 

638 
682 
726 

642 
686 
730 

647 
691 
734 

94 
95 
96 

739 

782 
826 

743 

787 

8:^ 

747 
791 
835 

752 
795 
839 

756 
800 
843 

760 
804 

848 

765 

808 
852 

769 
813 
856 

774 
817 
861 

778 
822 
865 

97 
98 
99 

870 
913 
957 

874 
917 
961 

878 
922 
965 

883 
926 
970 

887 
930 
974 

891 
935 
978 

896 
939 
983 

900 
944 

987 

904 
948 
991 

909 
952 
996 

1000 

00  000 

004 

009 

013 

.17 

022 

026 

030 

035 

039 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

20 


Logarithms  of  Important  Constants 


[la 


TABLE  la.     LOGARITHMS   OF  IMPORTANT  CONSTANTS 


i\r=  Number 


Value  of  JV 


LOGio  ^ 


"Vtt 
e  =  Napierian  Base 

M=logiQe 

l-4-3f=logel0 

180  -r-  IT  =  degrees  in  1  radian 

TT  -7- 180  =  radians  in  1° 

TT  -f- 10800  =  radians  in  1' 

TT  -4-  648000  =  radians  in  1" 

sin  1" 

tan  1" 

centimeters  in  1  ft. 

feet  in  1  cm. 

inches  in  1  m. 

pounds  in  1  kg. 

kilograms  in  1  lb. 

g  (average  value) 

weight  of  1  cu.  ft.  of  water 
weight  of  1  cu.  ft.  of  air 
cu.  m.  in  1  (U.  S.)  gallon 
ft.  lb.  per  sec.  in  1  H.  P. 
kg.  m.  per  sec.  in  1  H.  P. 
watts  in  1  H.  P. 


3.14159265 

0.31830989 

9.86960440 

1.77245385 

2.71828183 

0.43429448 

2.30258509 

57.2957795 

0.01745329 

0.0002908882 

0.000004848136811095 

0.000004848136811076 

0.000004848136811162 

30.480 

0.032808 

39.37 

2.20462 

0.453593 

32.16  ft./sec./sec. 

=  981  cm. /sec. /sec. 
62.425  lb.  (max.  density) 
0.0807  lb.  (at  32°  F.) 
231. 
550. 
76.0404 
745.957 


0.49714987 

9.5028,5013 

0.99429975 

0.24857494 

0.43429448 

9.63778431 

0.36221569 

1.75812263 

8.24187737 

().46372612 

4.68557487 

4.68557487 

4.68557487 

1.4840158 

8.5159842 

1.5951654 

0.3433340 

9.6566660  . 

1.5073 

2.9916690 

1.7953586 

8.907 

2.3636120 

2.7403627 

1.8810445 

2.8727135 


COMMON  LOGARITHMS  OF  THE  FIRST  HUNDRED  PRIME  NUMBERS 


N 

Logarithm 

N 

Log 

N 

Log 

N 

Log 

N 

Log 

1 

0000000000 

71 

8512583 

173 

2380461 

281 

4487063 

409 

6117233 

2 

3010299957 

73 

8633229 

179 

2528530 

283 

4517864 

419 

6222140 

3 

4771212547 

79 

8976271 

181 

2576786 

293 

4668676 

421 

6242821 

5 

6989700043 

83 

9190781 

191 

2810334 

307 

4871384 

431 

6344773 

7 

8450980400 

89 

9493900 

193 

2855573 

311 

4927604 

433 

6364879 

11 

0413926852 

97 

9867717 

197 

2944662 

313 

4955443 

439 

6424645 

13 

1139433523 

101 

0043214 

199 

2988531 

317 

5010593 

443 

6464037 

17 

2304489214 

103 

0128372 

211 

3242825 

331 

5198280 

449 

6522463 

19 

2787536010 

107 

0293838 

223 

3483049 

337 

5276299 

457 

6599162 

23 

3617278360 

109 

0374265 

227 

3560259 

347 

5403295 

461 

6637009 

29 

4623979979 

113 

0530784 

229 

3598355 

349 

5428254 

463 

6655810 

31 

4913616938 

127 

1038037 

233 

3673559 

353 

5477747 

467 

6693169 

37 

5682017241 

131 

1172713 

239 

3783979 

359 

5550944 

479 

6803355 

41 

6127838567 

137 

1367206 

241 

3820170 

367 

5646661 

487 

68752^)0 

43 

6334684556 

139 

1430148 

251 

3996737 

373 

5717088. 

491 

6910815 

47 

6720978579 

149 

1731863 

257 

4099331 

379 

5786392 

499 

6981005 

53 

7242758696 

151 

1789769 

263 

4199557 

383 

5831988 

503 

7015680 

59 

at 

7708520116 

TOKOOnO'JKA 

157 

1958997 

269 

071 

4297523 
A QonnoQ 

389 

rici'7 

5899496 

FC0C70nFC 

509 

K'}t 

7067178 

71A«^77 

TABLE   II 


ACTUAL  VALUES 


OF    THE 


TKIGONOMETEIC    FUNCTIONS 


FKOM 


0°  TO  90°  AT  INTERVALS   OF   ONE   MINUTE 


TO 


FIVE   DECIMAL   PLACES 


- 

III                       L 

1  LI.    I 

44-^                        sL 

u     91      41 

US                  il 

L                 4L     9a:     i 

U^                       n. 

r-_v        m     jz 

M^                    fi 

-         —          ^  rit  ^ 

'-J (2-                        Oj  ^-r^TlM   -  ^ 

/^                                Zf"      ^  eroed    ^./kj. 

/                                ."   r                         ^e    ^ 

AdT                            ^\    ^^t 

-        $-                        T     \^A 

f-                         ^'       t 

VST                          V  VI) 

->                          ^^~  "  "'                 M  AV 

/'       J 

-Ix                       ^^     ■                        0  lA. 

^o.               ^          J 

fS^                    ^^     1-       4^'^\-i^ 

^eear.-l'          ^^                                             ^J 

--%%   -                     ^.--T     ¥' 

^i"i-    -"                                                 ^/ 

^^c^V                -  ^^-'         "^*I    -/\i 

^^^"/■^^^                    -  -         _A>ir 

o<-         ^^                -rS^^ 

^          /                      !?»J^Vy,                                                  i,*^^ 

T'          ^N?*?               ^y 

-    '^pv          y^                                   T /^5S^ 

^  ^              "^^               / 

^-L                    ^. ^t 

■^              ^s   T            1- 

^_,.,.^.  .,  ,.                              ,^           P" 

\-o                         ^             7- 

-     ^   /           ^               .r       -^f?"' 

X5o^                               si-         j^      

i  -_4r  ^'^ .              "•.           ,^^  L'-c;\«^ 

>$<?                 ^^  /      i 

S^^                  _         _               ^yl___^ 

"y^       ""IT"                     ^?Ve^:^^*^    'tt 

iFi     T           .-,->J:        it 

I'^k   X               ^-.^  \       it 

77         antV                              4,^                si-^ 

5?                                                ^0-                  (^-45  X-5 

7::)-         C^O)       S                                 >/                      >^ 

r^.           ^*y                 X*?!^ 

^)  \        >                 rH 

'C^N     /                   \\/ 

:s>r:^'' .^b?'                 ± 

4^   S^                      vUj^^ 

_                 «6>\   7<.           c"^>>T 

r  J         ^'---^"^  ' 

ncliona                     '    ?v          ^  "^ "             "*  -J.  M 

QCiions                    L^.^^                       <^-L- 

Tses                      4/    r                          ^3X-- 

ises                        /     7  L                             1  ^i\ 

nrp  flin  <tr            /\     /     \                                   J     ^\ 

dD/^""  ^_             tit" 

zri         ^  ± 

TI7^.      t                      1 

it        "^  i 

lUc^                        t 

/           ^ 

7r/i^^n-                .      r 

22 


0°  — Values  of  Trigonometric  runctions— 1° 


/ 

Sin 

Tan 

Ctn 

Cos 

0 

.00000 

.00000 

1.0000 

60 

1 

029 

029 

3437.7 

000 

59 

2 

058 

058 

1718.9 

000 

58 

3 

087 

087 

1145.9 

000 

57 

4 

116 

116 

859.44 

000 

56 

5 

.00145 

.00145 

687.55 

1.0000 

55 

6 

175 

175 

572.96 

000 

54 

7 

204 

204 

491.11 

000 

53 

8 

233 

233 

429.72 

000 

52 

9 

262 

262 

381.97 

000 

51 

10 

.00291 

.00291 

343.77 

1.0000 

50 

11 

320 

320 

312.52 

.99999 

49 

12 

349 

349 

286.48 

999 

48 

13 

378 

378 

264.44 

999 

47 

14 

407 

407 

245.55 

999 

46 

15 

.00436 

.00436 

229.18 

.99999 

45 

1(5 

465 

465 

214.86 

999 

44 

17 

495 

495 

202.22 

999 

43 

18 

524 

524 

190.98 

999 

42 

19 

553 

553 

180.93 

998 

41 

20 

.00582 

.00582 

171.89 

.99998 

40 

21 

611 

611 

163.70 

998 

39 

22 

640 

640 

156.26 

998 

38 

23 

669 

669 

149.47 

998 

37 

24 

698 

698 

143.24 

998 

36 

25 

.00727 

.00727 

137.51 

.99997 

35 

26 

756 

756 

132.22 

997 

34 

27 

785 

785 

127.32 

997 

33 

28 

814 

815 

122.77 

997 

32 

29 

844 

844 

118.54 

996 

31 

30 

.00873 

.00873 

114.59 

.99996 

30 

31 

902 

902 

hO.89 

996 

29 

32 

931 

931 

107.43 

996 

28 

33 

960 

960 

104.17 

995 

27 

34 

.00989 

.00989 

101.11 

995 

26 

35 

.01018 

.01018 

98.218 

.99995 

25 

36 

047 

047 

95.489 

995 

24 

37 

076 

076 

92.908 

994 

23 

38 

105 

105 

90.463 

994 

22 

39 

134 

135 

88.144 

994 

21 

40 

.01164 

.01164 

85.940 

.99993 

20 

41 

193 

193 

83.844 

993 

19 

42 

222 

222 

81.847 

993 

18 

43 

251 

251 

79.943 

992 

17 

44 

280 

280 

78.126 

992 

16 

45 

.01309 

.01309 

76.390 

.99991 

15 

46 

338 

338 

74.729 

991 

14 

47 

367 

367 

73.139 

991 

13 

48 

396 

396 

71.615 

990 

12 

49 

425 

425 

70.153 

990 

11 

50 

.01454 

.01455 

68.750 

.99989 

10 

51 

483 

484 

67.402 

989 

9 

52 

513 

513 

66.105 

989 

8 

53 

542 

542 

64.858 

988 

7 

54 

571 

571 

63.657 

988 

6 

55 

.01600 

.01600 

62.499 

.99987 

6 

56 

629 

629 

61.383 

987 

4 

57 

658 

658 

60.306 

986 

3 

58 

687 

687 

59.266 

986 

2 

59 

716 

716 

68.261 

985 

1 

60 

.01745 

.01746 

57.290 

.99985 

0 

Cos 

Ctn 

Tan 

Sin 

f 

1 

Sin 

Tan 

Ctn 

Cos 

0 

.01745 

.01746 

57.290 

.99985 

60 

1 

774 

775 

56.351 

984 

59 

2 

803 

804 

55.442 

984 

58 

3 

832 

833 

54.561 

983 

57 

4 

862 

862 

53.709 

983 

56 

5 

.01891 

.01891 

52.882 

.99982 

55 

6 

920 

920 

52.081 

982 

54 

7 

949 

949 

51.303 

981 

53 

8 

.01978 

.01978 

50.549 

980 

52 

9 

.02007 

.02007 

49.816 

980 

51 

10 

.02036 

.02036 

49.104 

.99979 

50 

11 

065 

066 

48.412 

979 

49 

12 

094 

095 

47.740 

978 

48 

13 

123 

124 

47.085 

977 

47 

14 

152 

153 

46.449 

977 

46 

15 

.02181 

.02182 

45.829 

.99976 

45 

16 

211 

211 

45.226 

976 

44 

17 

240 

240 

44.639 

975 

43 

18 

269 

269 

44.066 

974 

42 

19 

298 

298 

43.508 

974 

41 

20 

.02327 

.02328 

42.964 

.99973 

40 

21 

356 

357 

42.433 

972 

39 

22 

385 

386 

41.916 

972 

38 

23 

414 

415 

41.411 

971 

37 

24 

443 

444 

40.917 

970 

36 

25 

.02472 

.02473 

40.436 

.99969 

35 

26 

501 

502 

39.965 

969 

34 

27 

530 

531 

39.506 

968 

33 

28 

660 

560 

39.057 

967 

32 

29 

589 

589 

38.618 

966 

31 

30 

.02618 

.02619 

38.188 

.99966 

30 

31 

647 

648 

37.769 

965 

29 

32 

676 

677 

37.358 

964 

28 

33 

705 

706 

36.956 

963 

27 

34 

734 

735 

36.563 

963 

26 

35 

.02763 

.02764 

36.178 

.99962 

25 

36 

792 

793 

35.801 

961 

24 

37 

821 

822 

35.431 

960 

23 

38 

850 

851 

35.070 

959 

22 

39 

879 

881 

34.715 

959 

21 

40 

.02908 

.02910 

34.368 

.99958 

20 

41 

938 

939 

34.027 

957 

19 

42 

967 

968 

33.694 

956 

18 

43 

.02996 

.02997 

33.366 

955 

17 

44 

.03025 

.03026 

33.045 

954 

16 

45 

.03054 

.03055 

32.730 

.99953 

15 

46 

083 

084 

32.421 

952 

14 

47 

112 

114 

32.118 

952 

13 

48 

141 

143 

31.821 

951 

12 

49 

170 

172 

31.528 

950 

11 

50 

.03199 

.03201 

31.242 

.99949 

10 

51 

228 

230 

30.960 

948 

9 

52 

257 

259 

30.683 

947 

8 

53 

286 

288 

30.412 

946 

7 

54 

316 

317 

30.145 

945 

6 

55 

.03345 

.03346 

29.882 

.99944 

5 

56 

374 

376 

29.624 

943 

4 

57 

403 

405 

29.371 

942 

3 

58 

432 

434 

29.122 

941 

2 

59 

461 

463 

28.877 

940 

1 

60 

.03490 

.03492 

28.636 

.99939 

0 

Cos 

Ctn 

Tan 

Sin   '  1 

iq 

2°— Values  of  Trigonometric  Functions  — 3 

0 

23 

/ 

Sin 

Tan 

Ctn 

Cos 

f 

Sin 

Tan 

ctn 

Cos 

0 

.03490 

.03492 

28.636 

.99939 

60 

0 

.05234 

.05241 

19.081 

.99863 

60 

1 

519 

521 

.399 

938 

59 

1 

263 

270 

18.976 

861 

59 

2 

548 

550 

28.166 

937 

58 

2 

292 

299 

.871 

860 

58 

.  3 

577 

579 

27.937 

936 

67 

3 

321 

328 

.768 

858 

57 

4 

606 

609 

.712 

935 

56 

4 

350 

357 

.666 

857 

56 

5 

.03635 

.03638 

27.490 

.99934 

55 

5 

.05379 

.05387 

18.564 

.99855 

55 

664 

667 

.271 

933 

54 

6 

408 

416 

..464 

854 

54 

693 

696 

27.057 

932 

63 

7 

437 

445 

.366 

852 

53 

'.   o 

723 

725 

26.845 

931 

52 

8 

466 

474 

.268 

851 

52 

1 '' 

752 

754 

.637 

930 

51 

9 

495 

503 

.171 

849 

51 

lio 

.03781 

.03783 

26.432 

.99929 

50 

10 

.05524 

.05533 

18075 

.99847 

50 

111 

810 

812 

.230 

927 

49 

11 

.553 

562 

17.980 

846 

49 

'12 

839 

842 

26.031 

926 

48 

12 

582 

591 

.886 

844 

48 

(13 

868 

871 

25.836 

925 

47 

13 

611 

620 

.793 

842 

47 

;i4 

897 

900 

.642 

924 

46 

14 

640 

649 

.702 

841 

46 

15 

.03926 

.03929 

25.452 

.99923 

45 

15 

.05669 

.05678 

17.611 

.99839 

45 

W 

955 

958 

.264 

922 

44 

16 

698 

708 

.521 

838 

44 

17 

.03984 

.03987 

25.080 

921 

43 

17 

727 

737 

.431 

836 

43 

18 

.04013 

.04016 

24.898 

919 

42 

18 

756 

766 

.343 

834 

42 

19 

042 

046 

.719 

918 

41 

19 

786 

795 

.256 

833 

41 

20 

.04071 

.04075 

24.542 

.99917 

40 

20 

.06814 

.05824 

17.169 

.99831 

40 

21 

100 

104 

.368 

916 

39 

21 

844 

854 

17.084 

829 

39 

22 

129 

133 

.UX) 

915 

38 

22 

873 

883 

16.999 

827 

38 

23 

169 

162 

24.026 

913 

37 

23 

902 

912 

.915 

826 

37 

24 

188 

191 

23.859 

912 

36 

24 

931 

941 

.832 

824 

36 

25 

.04217 

.04220 

23.695 

.99911 

35 

25 

.05960 

.05970 

16.750 

.99822 

35 

2(3 

24(5 

250 

.532 

910 

34 

26 

.05989 

.05999 

.668 

821 

34 

27 

275 

279 

.372 

909 

33 

27 

.06018 

.06029 

.587 

819 

33 

28 

304 

308 

.214 

907 

32 

28 

047 

058 

.507 

817 

32 

29 

333 

337 

23.058 

906 

31 

29 

076 

087 

.428 

815 

31 

30 

.04362 

.04366 

22.904 

.99905 

30 

30 

.06105 

.06116 

16.350 

.99813 

30 

31 

391 

395 

.752 

904 

29 

31 

134 

145 

.272 

812 

29 

32 

420 

424 

.602 

902 

28 

32 

163 

175 

.195 

810 

28 

33 

449 

454 

.454 

901 

27 

33 

192 

204 

.119 

808 

27 

34 

478 

48§ 

.308 

900 

26 

34 

221 

233 

16.043 

806 

26 

35 

.04507 

.04512 

22.164 

.99898 

25 

35 

.06250 

.06262 

15.969 

.99804 

25 

36 

536 

541 

22.022 

897 

24 

36 

279 

291 

.895 

803 

24 

37 

565 

570 

21.881 

896 

23 

37 

308 

321 

.821 

801 

23 

38 

594 

699 

.743 

894 

22 

38 

337 

350 

.748 

799 

22 

39 

623 

628 

.606 

893 

21 

39 

366 

379 

.676 

797 

21 

40 

.04653 

.04658 

21.470 

.99892 

20 

40 

.06395 

.06408 

16.605 

.99796 

20 

41 

682 

687 

.337 

8^)0 

19 

41 

424 

438 

.534 

793 

19 

42 

711 

716 

.205 

889 

18 

42 

453 

467 

.464 

792 

18 

43 

740 

745 

21.075 

888 

17 

43 

482 

496 

.394 

790 

17 

44 

769 

774 

20.946 

886 

16 

44 

511 

525 

.326 

788 

m 

45 

.04798 

.04803 

20.819 

.99885 

15 

45 

.06540 

.06554 

16.257 

.99786 

15 

46 

827 

833 

.693 

883 

14 

46 

569 

584 

.189 

784 

14 

47 

856 

862 

.569 

882 

13 

47 

598 

613 

.122 

782 

13 

48 

885 

891 

.446 

881 

12 

48 

627 

642 

16.056 

780 

12 

49 

914 

920 

.325 

879 

11 

49 

656 

671 

14.990 

778 

11 

50 

.04943 

.04949 

20.206 

.99878 

10 

50 

.06685 

.06700 

14.924 

.99776 

10 

51 

.04972 

.04978 

20.087 

876 

9 

51 

714 

730 

.860 

774 

9 

52 

.05001 

.05007 

19.970 

875 

8 

52 

743 

759 

.795 

772 

8 

53 

030 

037 

.855 

873 

7 

53 

773 

788 

.732 

770 

7 

54 

059 

066 

.740 

872 

6 

54 

802 

817 

.669 

768 

6 

55 

.05088 

.05095 

19.627 

.99870 

5 

55 

.06831 

.06847 

14.606 

.99766 

5 

56 

117 

124 

.516 

869 

4 

56 

860 

876 

.644 

764 

4 

57 

146 

153 

.406 

867 

3 

57 

889 

905 

.482 

762 

3 

58 

175 

182 

.296 

866 

2 

58 

918 

934 

.421 

760 

2 

59 

205 

212 

.188 

864 

1 

59 

947 

963 

.361 

758 

1 

60 

.05234 

.05241 

19.081 

.99863 

0 

60 

.06976 

.06993 

14.301 

.99756 

0 

rina 

n+n 

Tqti 

SlTl 

1 

n.na 

ntn 

Tart 

Sin 

/ 

24 


4°— Values  of  Trigonometric  Functions— 5° 


[n 


/ 

Sin 

Tan 

Ctn 

Cos 

0 

.06976 

.06993 

14.301 

.99756 

60 

1 

.07005 

.07022 

.241 

754 

59 

2 

034 

051 

.182 

752 

58 

3 

063 

080 

.124 

750 

57 

4 

092 

]10 

.065 

748 

56 

5 

.07121 

.07139 

14.008 

.99746 

55 

6 

150 

168 

13.951 

744 

54 

7 

179 

197 

.894 

742 

53 

8 

208 

227 

.838 

740 

52 

9 

237 

256 

.782 

738 

51 

10 

.07266 

.07285 

13.727 

.99736 

50 

11 

295 

314 

•  .672 

734 

49 

12 

324 

344 

.617 

731 

48 

13 

353 

373 

.563 

729 

47 

14 

382 

402 

.510 

727 

46 

15 

.07411 

.07431 

13.457 

.99725 

45 

16 

440 

461 

.404 

723 

44 

17 

469 

490 

.352 

721 

43 

18 

498 

519 

.300 

719 

42 

19 

527 

548 

.248 

716 

41 

20 

.07556 

.07578 

13.197 

.99714 

40 

21 

585 

607 

.146 

712 

39 

22 

614 

636 

.096 

710 

38 

23 

643 

665 

13.046 

708 

37 

24 

672 

695 

12.996 

705 

36 

25 

.07701 

.07724 

12.947 

.99703 

35 

26 

730 

753 

.898 

701 

34 

27 

759 

782 

.850 

699 

33 

28 

788 

812 

.801 

696 

32 

29 

817 

841 

.754 

694 

31 

30 

.07846 

.07870 

12.706 

.99692 

30 

31 

875 

899 

.659 

689 

29 

32 

904 

929 

.612 

687 

28 

33 

933 

958 

.566 

685 

27 

34 

962 

.07987 

.520 

683 

26 

35 

.07991 

.08017 

12.474 

.99680 

25 

36 

.08020 

046 

.429 

678 

24 

37 

049 

075 

.384 

676 

23 

38 

078 

104 

.339 

673 

22 

39 

107 

134 

.295 

671 

21 

40 

.08136 

.08163 

12.251 

.99668 

20 

41 

165 

192 

.207 

666 

19 

42 

194 

221 

.163 

664 

18 

43 

223 

251 

.120 

661 

17 

44 

252 

280 

.077 

659 

16 

45 

.08281 

.08309 

12.035 

.99657 

16 

46 

310 

339 

11.992 

654 

14 

47 

339 

368 

.950 

652 

13 

48 

368 

397 

.909 

649 

12 

49 

397 

427 

.867 

647 

11 

50 

.08426 

.08456 

11.826 

.99644 

10 

51 

455 

485 

.785 

642 

9 

52 

484 

514 

.745 

639 

8 

53 

513 

544 

.705 

637 

7 

54 

542 

573 

.664 

635 

6 

55 

.08571 

.08602 

11.625 

.99632 

5 

56 

600 

632 

.585 

630 

4 

57 

629 

661 

.546 

627 

3 

58 

658 

690 

.507 

625 

2 

59 

687 

720 

.468 

622 

1 

60 

.08716 

.08749 

11.430 

.99619 

0 

Cos 

Ctn 

Tan 

Sin 

1 

f 

Sin 

Tan 

Ctn 

Cos 

0 

.08716 

.08749 

11.430 

.99619 

60 

1 

745 

778 

.392 

617 

2 

774 

807 

.354 

614 

5^ 

3 

803 

837 

.316 

612 

5/^ 

4 

831 

866 

.279 

609 

m 

5 

.08860 

.08895 

11.242 

.99607 

55 

6 

889 

925 

.205 

604 

64  ' 

7 

918 

954 

.168 

602 

53 

8 

947 

.08983 

.132 

699 

52  >, 

9 

.08976 

.09013 

.095 

696 

51  ] 

10 

.09005 

.09042 

11.059 

.99594 

50  ' 

11 

034 

071 

11.024 

591 

49) 

12 

063 

101 

10.988 

588 

48 

13 

092 

130 

.953 

586 

47! 

14 

121 

159 

.918 

583 

46 

15 

.09150 

.09189 

10.883 

.99580 

45 

16 

179 

218 

.848 

578 

44 

17 

208 

247 

.814 

675 

43. 

18 

237 

277 

.780 

672 

42. 

19 

266 

306 

.746 

570 

41 

20 

.09295 

.09335 

10.712 

.99567 

40 

21 

324 

365 

.678 

564 

39 

22 

353 

394 

.645 

562 

38 

23 

382 

423 

.612 

559 

37 

24 

411 

453 

.579 

556 

36 

25 

.09440 

.09482 

10.546 

.99553 

35 

26 

469 

511 

.514 

551 

34 

27 

498 

541 

.481 

648 

33 

28 

527 

570 

.449 

545 

32 

29 

556 

600 

.417 

642 

31 

30 

.09585 

.09629 

10.385 

.99540 

30 

31 

614 

658 

.354 

637 

29 

32 

642 

688 

.322 

534 

28 

33 

671 

717 

.291 

531 

27 

34 

700 

746 

•  .260 

528 

26 

35 

.09729 

.09776 

10.229 

.99526 

25 

36 

758 

805 

.199 

523 

24 

37 

787 

834 

.168 

520 

23 

38 

816 

864 

.138 

517 

22 

39 

845 

893 

.108 

514 

21 

40 

.09874 

.09923 

10.078 

.99511 

20 

41 

903 

952 

.048 

508 

19 

42 

932 

.09981 

10.019 

506 

18 

43 

961 

.10011 

9.9893 

503 

17 

44 

.09990 

040 

.9601 

600 

16 

45 

.10019 

.10069 

9.9310 

.99497 

15 

46 

048 

099 

.9021 

494 

14 

47 

077 

128 

.8734 

491 

13 

48 

106 

158 

.8448 

488 

12 

49 

135 

187 

.8164 

485 

11 

50 

.10164 

.10216 

9.7882 

.99482 

10 

51 

192 

246 

.7601 

479 

9 

52 

221 

275 

.7322 

476 

8 

53 

250 

305 

.7044 

473 

7 

54 

279 

334 

.6768 

470 

6 

55 

.10308 

.10363 

9.6493 

.99467 

5 

56 

337 

393 

.6220 

464 

4 

57 

366 

422 

.5949 

461 

3 

58 

395 

452 

.5679 

458 

2 

59 

424 

481 

.5411 

455 

1 

60 

.10453 

.10510 

9.5144 

.99452 

0 

Cos 

Ctn 

Tan 

Sin 

1 

IIJ 


6°— Values  of  Trigonometric  Functions  — 7° 


4 
5 

6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
1(J 
17 
18 
19 
20 
21 
22 
23 
24 
26 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


Sin 


.10453 
482 
511 
540 
569 

.10597 
626 
655 
684 
713 

.10742 
771 
800 
829 
858 

.10887 
916 
945 

.10973 

.11002 

.11031 
060 
089 
118 
147 

.11176 
205 
234 
263 
291 

.11320 
349 
378 
407 
436 

.11465 
494 
523 
552 
580 

.11609 
638 
667 
696 
725 

.11754 
783 
812 
840 
869 

.11898 
927 
956 

.11985 

.12014 

.12043 
071 
100 
129 
158 

.12187 


Tan   Gtn   Cos 


.10510 
540 
569 
599 
628 

.10657 
687 
716 
746 
775 

.10805 
834 
863 
893 
922 

.10952 

.10981 

.11011 
040 
070 

.11099 
128 
158 
187 
217 

.11246 
276 
305 
335 
364 

.11394 
423 
452 
482 
511 

.11541 
570 
600 
629 
659 

.11688 
718 
747 
777 
806 

.11836 
865 
895 
924 
954 

.11983 

.12013 
042 
072 
101 

.12131 
160 
190 
219 
249 

.12278 


9.5144 
.4878 
.4614 
.4352 
.4090 

9.3831 
.3572 
.3315 
.30(^0 
.2806 

9.2553 
.2302 
.2052 
.1803 
.1555 

9.1309 
.1065 
.0821 
.0579 
.0338 

9.0098 

8.9860 
.9623 
.9387 
.9152 

8.8919 
.8686 
.8455 
.8225 
.7996 

8.7769 
.7542 
.7317 
.7093 
.6870 

8.6648 
.(;427 
.6208 
.5989 
.5772 

8.5555 
.5340 
.5126 
.4913 
.4701 

8.4490 
.4280 
.4071 
.3863 
.3656 

8.3450 
.3245 
.3041 
.2838 
.2636 

8.2434 
.2234 
.2035 
.1837 
.1640 

8.1443 


Cos   Ctn  I  Tan   Sin   ' 


.99452 
449 
446 
443 
440 

.99437 
434 
431 
428 
424 

.99421 
418 
415 
412 
409 

.99406 
402 
399 
396 
393 

.99390 
386 
383 
380 
377 

.99374 
370 
367 
364 
360 

.99357 
354 
351 
317 
344 

.99341 
337 
334 
331 
327 

.99324 
320 
317 
314 
310 

.99307 
303 
300 
297 
293 

.99290 
286 
283 
279 
276 

.99272 
269 
265 
262 
258 

.99255 


/ 

Sin 

Tan 

Ctn 

Cos 

0 

.12187 

.12278 

8.1443 

.99255 

60 

1 

216 

308 

.1248 

251 

59 

2 

245 

338 

.1054 

248 

58 

3 

274 

367 

.0860 

244 

57 

4 

302 

397 

.0667 

240 

56 

5 

.12331 

.12426 

8.0476 

.99237 

65 

6 

360 

456 

.0285 

233 

54 

7 

389 

485 

8.0096 

230 

53 

8 

418 

515 

7.9906 

226 

52 

9 

447 

544 

.9718 

222 

51 

10 

.12476 

.12574 

7.9530 

.99219 

50 

11 

504 

603 

.9344 

215 

49 

12 

533 

633 

.9158 

211 

48 

13 

562 

662 

.8973 

208 

47 

14 

591 

692 

.8789 

204 

46 

15 

.12620 

.12722 

7.8606 

.99200 

45 

16 

<;49 

751 

.8424 

197 

44 

17 

()78 

781 

.8243 

193 

43 

18 

706 

810 

.8062 

189 

42 

19 

735 

840 

.7882 

186 

41 

20 

.12764 

.12869 

7.7704 

.99182 

40 

21 

793 

899 

.7525 

178 

39 

22 

822 

929 

.7348 

175 

38 

23 

851 

958 

.7171 

171 

37 

24 

880 

.12988 

.6996 

167 

36 

25 

.12908 

.13017 

7.6821 

.99163 

35 

26 

937 

047 

.6647 

160 

34 

27 

9()6 

076 

.6473 

156 

33 

28 

.12^)95 

106 

.6301 

152 

32 

29 

.13024 

136 

.6129 

148 

31 

30 

.13053 

.13165 

7.5958 

.99144 

30 

31 

081 

195 

.5787 

141 

29 

32 

110 

224 

.5618 

137 

28 

33 

139 

254 

.5449 

133 

27 

34 

168 

284 

.5281 

129 

2(> 

35 

.13197 

.13313 

7.5113 

.99125 

25 

36 

226 

343 

.4947 

122 

24 

37 

254 

372 

.4781 

118 

23 

38 

283 

402 

.4615 

114 

22 

39 

312 

432 

.4451 

110 

21 

40 

.13341 

.13461 

7.4287 

.99106 

20 

41 

370 

491 

.4124 

102 

19 

42 

399 

521 

.3962 

098 

18 

43 

427 

550 

.3800 

094 

17 

44 

456 

580 

.3639 

091 

16 

45 

.13485 

.13609 

7.3479 

.99087 

15 

4() 

514 

639 

.3319 

083 

14 

47 

543 

669 

.3160 

079 

13 

48 

572 

698 

.3002 

075 

12 

49 

600 

728 

.2844 

071 

11 

60 

.13629 

.13758 

7.2687 

.99067 

10 

51 

658 

787 

.2531 

063 

9 

52 

687 

817 

.2375 

059 

8 

53 

716 

846 

.2220 

055 

7 

54 

744 

876 

.2066 

051 

6 

55 

.13773 

.13906 

7.1912 

.99047 

5 

56 

802 

935 

.1759 

043 

4 

57 

831 

965 

.1607 

039 

3 

58 

860 

.13995 

.1455 

035 

2 

59 

889 

.14024 

.1304 

031 

1 

60 

.13917 

.14054 

7.1154 

.99027 

0 

Cos 

Ctn 

Tan 

Sin 

1 

26 


8°  — Values  of  Trigonometric  Functions— 9° 


/ 

Sin 

Tan 

Ctn 

Cos 

0 

.13917 

.14054 

7.1154 

.99027 

60 

1 

946 

084 

.1004 

023 

59 

2 

.13975 

113 

.0855 

019 

58 

3 

.14004 

143 

.0706 

015 

57 

4 

033 

173 

.0558 

Oil 

56 

5 

.14061 

.14202 

7.0410 

.99006 

55 

6 

090 

232 

.0264 

.99002 

54 

7 

119 

262 

7.0117 

.98998 

53 

8 

148 

291 

6.9972 

994 

52 

9 

177 

321 

.9827 

990 

51 

10 

.14205 

.14351 

6.9682 

.98986 

50 

11 

234 

381 

.9538 

982 

49 

12 

263 

410 

.9395 

978 

48 

13 

292 

440 

.9252 

973 

47 

14 

320 

470 

.9110 

969 

46 

15 

.14349 

.14499 

6.8969 

.98965 

45 

16 

378 

529 

.8828 

961 

44 

17 

407 

559 

.8687 

957 

43 

18 

436 

588 

.8548 

953 

42 

19 

464 

618 

.8408 

948 

41 

20 

.14493 

.14648 

6.8269 

.98944 

40 

21 

522 

678 

.8131 

940 

39 

22 

551 

707 

.7994 

936 

38 

23 

580 

737 

.7856 

931 

37 

24 

608 

767 

.7720 

927 

36 

25 

.14637 

.14796 

6.7584 

.98923 

35 

20 

em 

826 

.7448 

919 

34 

27 

695 

856 

.7313 

914 

33 

28 

723 

886 

.7179 

910 

3*2 

29 

752 

915 

.7045 

906 

31 

30 

.14781 

.14945 

6.6912 

.98902 

30 

31 

810 

.14975 

.6779 

897 

29 

32 

838 

.15005 

.6646 

893 

28 

33 

867 

034 

.6514 

889 

27 

34 

896 

064 

.6383 

884 

26 

35 

.14925 

.15094 

6.6252 

.98880 

25 

36 

954 

124 

.6122 

876 

24 

37 

.14982 

153 

.5992 

871 

23 

38 

.15011 

183 

.5863 

867 

22 

39 

040 

213 

.5734 

863 

21 

40 

.15069 

.15243 

6.5606 

.98858 

20 

41 

097 

272 

.5478 

854 

19 

42 

126 

302 

.5350 

849 

18 

43 

155 

332 

.5223 

845 

17 

44 

184 

362 

.5097 

841 

16 

45 

.15212 

.15391 

6.4971 

.98836 

15 

46 

241 

421 

.4846 

832 

14 

47 

270 

451 

4721 

827 

13 

48 

299 

481 

.4596 

823 

12 

49 

327 

511 

.4472 

818 

11 

50 

.15356 

.15540 

6.4348 

.98814 

10 

51 

.  385 

570 

.4225 

809 

9 

52 

414 

600 

.4103 

805 

8 

53 

442 

630 

.3980 

800 

7 

54 

471 

660 

.3859 

796 

6 

55 

.15500 

.15689 

6.3737 

.98791 

5 

66 

529 

719 

.3617 

787 

4 

57 

557 

749 

.3496 

782 

3 

58 

586 

779 

.3376 

778 

2 

59 

615 

809 

.3257 

773 

1 

60 

.15643 

.15838 

6.3138 

.98769 

0 

Cos 

Ctn 

Tan 

Sin 

1 

/ 

Sin 

Tan 

Ctn 

Cos 

0 

.15643 

.15838 

6.3138 

.98769 

60 

1 

672 

868 

.3019 

764 

59 

2 

701 

898 

.2901 

760 

58 

3 

730 

928 

.2783 

755 

57 

4 

758 

958 

.2666 

751 

56 

6 

.15787 

.15988 

6.2549 

.98746 

65 

6 

816 

.16017 

.2432 

741 

54 

7 

845 

047 

.2316 

737 

53 

8 

873 

077 

.2200 

732 

52 

9 

902 

107 

.2085 

728 

51 

10 

.15931 

.16137 

6.1970 

.98723 

60 

11 

959 

167 

.1856 

718 

49 

12 

.15988 

196 

.1742 

714 

48 

13 

.16017 

226 

.1628 

709 

47 

14 

046 

256 

.1515 

704 

46 

15 

.16074 

.16286 

6.1402 

.98700 

45 

16 

103 

316 

.1290 

695 

44 

17 

132 

346 

.1178 

690 

43 

18 

160 

376 

.1066 

686 

42 

19 

189 

405 

.0955 

681 

41 

20 

.16218 

.16435 

6.0844 

.98676 

40 

21 

246 

465 

.0734 

671 

39 

22 

275 

495 

.0624 

667 

38 

23 

304 

525 

.0514 

662 

37 

24 

333 

555 

.0405 

657 

36 

25 

.16361 

.16585 

6.0296 

„98652 

36 

26 

390 

615 

.0188 

648 

34 

27 

419 

645 

6.0080 

643 

33 

28 

447 

674 

5.9972 

638 

32 

29 

476 

704 

.9865 

633 

31 

30 

.16505 

.16734 

5.9758 

.98629 

30 

31 

533 

764 

.9651 

624 

29 

32 

562 

794 

.9545 

619 

28 

33 

591 

824 

.9439 

614 

27 

34 

620 

854 

.9333 

609 

26 

35 

.16648 

.16884 

5.9228 

.98604 

26 

36 

677 

914 

.9124 

600 

24 

37 

706 

944 

.9019 

595 

23 

38 

734 

.16974 

.8915 

590 

22 

39 

763 

.17004 

.8811 

585 

21 

40 

.16792 

.17033 

5.8708 

.98580 

20 

41 

820 

063 

.8605 

575 

19 

42 

849 

093 

.8502 

570 

18 

43 

878 

123 

.8400 

565 

17 

44 

906 

153 

.8298 

561 

16 

45 

.16935 

.17183 

5.8197 

.98556 

16 

46 

964 

213 

.8095 

551 

14 

47 

.16992 

243 

.7994 

546 

13 

48 

.17021 

273 

.7894 

541 

12 

49 

050 

303 

.7794 

536 

11 

50 

.17078 

.17333 

5.7694 

.98531 

10 

51 

107 

363 

.7594 

526 

9 

52 

136 

393 

.7495 

521 

8 

53 

164 

423 

.7396 

516 

7 

54 

193 

453 

.7297 

511 

6 

65 

.17222 

.17483 

5.7199 

.98506 

5 

56 

250 

513 

.7101 

501 

4 

57 

279 

543 

.7004 

496 

3 

58 

308 

573 

.6906 

491 

2 

59 

336 

603 

.6809 

486 

1 

60 

.17365 

.17633 

5.6713 

.98481 

0 

Cos 

Ctn 

Tan 

Sin 

II] 


10°— Values  of  Trigonometric  Functions  — IF 


27 


f 

Sin 

Tan 

Ctn 

Cos 

0 

.17365 

.17633 

5.6713 

.98481 

60 

1 

393 

663 

.6617 

476 

59 

2 

422 

693 

-.6521 

471 

58 

3 

451 

723 

.6425 

466 

57 

4 

479 

753 

.6329 

461 

56 

5 

.17508 

.17783 

5.6234 

.98455 

55 

G 

537 

813 

.6140 

450 

54 

7 

565 

843 

.6045 

445 

53 

8 

594 

873 

.5951 

440 

52 

9 

623 

903 

.5857 

435 

51 

10 

.17651 

.17933 

5.5764 

.98430 

50 

11 

680 

963 

.5671 

425 

49 

12 

708 

.17993 

.5578 

420 

48 

13 

737 

.18023 

.5485 

414 

47 

14 

766 

053 

.5393 

409 

46 

15 

.17794 

.18083 

5.5301 

.98404 

45 

16 

823 

113 

.5209 

399 

44 

17 

852 

143 

.5118 

394 

43 

18 

880 

173 

.5026 

389 

42 

19 

909 

203 

.4936 

383 

41 

20 

.17937 

.18233 

5.4845 

.98378 

40 

21 

966 

263 

.4755 

373 

39 

22 

.17995 

293 

.4665 

368 

38 

23 

.18023 

323 

.4575 

362 

37 

24 

052 

353 

.4486 

357 

36 

25 

.18081 

.18384 

5.4397 

.98352 

35 

26 

109 

414 

.4308 

347 

34 

27 

138 

444 

.4219 

341 

33 

28 

166 

474 

.4131 

336 

32 

29 

195 

504 

.4043 

331 

31 

30 

.18224 

.18534 

5.3955 

.98325 

30 

31 

252 

564 

.3868 

320 

29 

32 

281 

594 

.3781 

315 

28 

33 

309 

624 

.3694 

310 

27 

34 

338 

654 

.3607 

304 

26 

35 

.18367 

.18684 

5.3521 

.98299 

25 

36 

395 

714 

.3435 

294 

24 

37 

424 

745 

.3349 

288 

28 

38 

452 

775 

.3263 

283 

22 

39 

481 

805 

.3178 

277 

21 

40 

.18509 

.18835 

5.3093 

.98272 

20 

41 

538 

865 

.3008 

267 

19 

42 

567 

895 

.2924 

261 

18 

43 

595 

925 

.2839 

256 

17 

44 

624 

955 

.2755 

250 

16 

45 

.18652 

.18986 

5.2672 

.98245 

15 

46 

681 

.19016 

.2588 

240 

14 

47 

710 

046 

.2505 

234 

13 

48 

738 

076 

.2422 

229 

12 

49 

767 

106 

.2339 

223 

11 

50 

.18795 

.19136 

5.2257 

.98218 

10 

51 

824 

166 

.2174 

212 

9 

52 

852 

197 

.2092 

207 

8 

53 

881 

227 

.2011 

201 

7 

54 

910 

257 

.1929 

196 

6 

55 

.18938 

.19287 

5.1848 

.98190 

5 

56 

'   967 

317 

.1767 

185 

4 

57 

.18995 

347 

.1686 

179 

3 

58 

.19024 

378 

.1606 

174 

2 

59 

052 

408 

.1526 

168 

1 

60 

.19081 

.19438 

5.1446 

.98163 

0 

Cos 

Ctn 

Tan 

Sin 

/ 

f 

Sin 

Tan 

Ctn 

Cos 

0 

.19081 

.19438 

5.1446 

.98163 

60 

1 

109 

468 

.1366 

157 

59 

2 

138 

498 

.1286 

152 

58 

3 

167 

529 

.1207 

146 

57 

4 

195 

559 

.1128 

140 

56 

5 

.19224 

.19589 

5.1049 

.98135 

55 

6 

252 

619 

.0970 

129 

54 

7 

281 

649 

.0892 

124 

53 

8 

309 

680 

.0814 

118 

52 

9 

338 

710 

.0736 

112 

51 

10 

.19366 

.19740 

5.0658 

.98107 

50 

11 

395 

770 

.0581 

101 

49 

12 

423 

801 

.0504 

09(5 

48 

13 

452 

831 

.0427 

090 

47 

14 

481 

861 

.0350 

084 

46 

15 

.19509 

.19891 

5.0273 

.98079 

45 

16 

538 

921 

.0197 

073 

44 

17 

566 

952 

.0121 

067 

43 

18 

595 

.19982 

5.0045 

061 

42 

19 

623 

.20012 

4.9969 

056 

41 

20 

.19652 

.20042 

4.9894 

.98050 

40 

21 

680 

073 

.9819 

044 

39 

22 

709 

103 

.9744 

039 

38 

23 

737 

133 

.9669 

033 

37 

24 

766 

164 

.9594 

027 

36 

25 

.19794 

.20194 

4.9520 

.98021 

35 

26 

823 

224 

.9446 

016 

34 

27 

851 

254 

.9372 

010 

33 

28 

880 

285 

.9298 

.98004 

32 

29 

908 

315 

.9225 

.97998 

31 

30 

.19937 

.20345 

4.91.52 

.97992 

30 

31 

965 

376 

.9078 

987 

29 

32 

.19994 

406 

.9006 

981 

28 

33 

.20022 

436 

.8933 

975 

27 

34 

051 

466 

.8860 

969 

2(; 

35 

.20079 

.20497 

4.8788 

.97963 

25 

36 

108 

527 

.8716 

958 

24 

37 

136 

557 

.8644 

952 

23 

38 

165 

588 

.8573 

946 

22 

39 

193 

618 

.8501 

940 

21 

40 

.20222 

.20648 

4.8430 

.97934 

20 

41 

250 

679 

.8359 

928 

19 

42 

279 

709 

.8288 

922 

18 

43 

307 

739 

.8218 

916 

17 

44 

336 

770 

.8147 

910 

16 

45 

.20364 

.20800 

4.8077 

.97905 

15 

46 

393 

830 

.8007 

899 

14 

47 

421 

861 

.7937 

893 

13 

48 

450 

891 

.7867 

887 

12 

49 

478 

921 

.7798 

881 

11 

50 

.20507 

.20952 

4.7729 

.97875 

10 

51 

535 

.20982 

.7659 

869 

9 

52 

563 

.21013 

.7591 

863 

8 

53 

592 

043 

.7522 

857 

7 

54 

620 

073 

.7453 

851 

6 

55 

.20649 

.21104 

4.7385 

.97845 

5 

56 

677 

134 

.7317 

839 

4 

57 

706 

164 

.7249 

833 

3 

58 

734 

195 

.7181 

827 

2 

59 

763 

225 

.7114 

821 

1 

60 

.20791 

.21256 

4.7046 

.97815 

0 

Cos 

Ctn 

Tan 

Sin 

/ 

28 

12°  — Values  of  Trigonometi 

'ic  Functions  — 13^ 

[n 

t 

Sin 

Tan 

Ctn 

Cos 

1 

Sin 

Tan 

Ctn 

Cos 

0 

.20791 

.21256 

4.7046 

.97815 

60 

0 

.22495 

.23087 

4.3315 

.97437 

60 

1 

820 

286 

.6979 

809 

59 

1 

523 

117 

.3257 

430 

59 

2 

848 

316 

.6912 

803 

58 

2 

552 

148 

.3200 

424 

58 

3 

877 

347 

.6845 

797 

57 

3 

580 

179 

.3143 

417 

57 

4 

905 

377 

.6779 

791 

56 

4 

608 

209 

.3086 

411 

56 

5 

.20933 

.21408 

4.6712 

.97784 

55 

5 

.22637 

.23240 

4.3029 

.97404 

55 

6 

962 

438 

.6646 

778 

54 

6 

665 

271 

.2972 

398 

54 

7 

.20990 

469 

.6580 

772 

53 

7 

693 

301 

.2916 

391 

53 

8 

.21019 

499 

.6514 

766 

52 

8 

722 

332 

.2859 

384 

52 

9 

047 

529 

.6448 

760 

51 

9 

750 

363 

.2803 

378 

51 

10 

.21076 

.21560 

4.6382 

.97754 

50 

10 

.22778 

.23393 

4.2747 

.97371 

60 

11 

104 

590 

.6317 

748 

49 

11 

807 

424 

.2691 

365 

49 

12 

132 

621 

.6252 

742 

48 

12 

835 

455 

.2635 

358 

48 

13 

161 

651 

.6187 

735 

47 

13 

863 

485 

.2580 

351 

47 

14 

189 

682 

.6122 

729 

46 

14 

892 

616 

.2524 

345 

46 

15 

.21218 

.21712 

4.6057 

.97723 

45 

15 

.22920 

.23547 

4.2468 

.97338 

45 

16 

246 

743 

.5993 

717 

44 

16 

948 

578 

.2413 

331 

44 

17 

275 

773 

.5928 

711 

43 

17 

.22977 

608 

.2358 

325 

43 

18 

303 

804 

.5864 

705 

42 

18 

.23005 

639 

.2303 

318 

42 

19 

331 

834 

.5800 

698 

41 

19 

033 

670 

.2248 

311 

41 

20 

.21360 

.21864 

4.5736 

.97692 

40 

20 

.23062 

.23700 

4.2193 

.97304 

40 

21 

388 

895 

.5673 

686 

39 

21 

090 

731 

.2139 

298 

39 

22 

417 

925 

.5609 

680 

38 

22 

118 

762 

.2084 

291 

38 

23 

445 

956 

.5546 

673 

37 

23 

146 

793 

.2030 

284 

37 

24 

474 

.21986 

.5483 

667 

36 

24 

175 

823 

.1976 

278 

36 

25 

.21502 

.22017 

4.5420 

.97661 

35 

25 

.23203 

.23854 

4.1922 

.97271 

35 

26 

530 

047 

.5357 

655 

34 

26 

231 

885 

.1868 

264 

34 

27 

559 

078 

.5294 

648 

33 

27 

260 

916 

.1814 

257 

33 

28 

587 

108 

.5232 

642 

32 

28 

288 

946 

.1760 

251 

32 

29 

616 

139 

.5169 

636 

31 

29 

316 

.23977 

.1706 

244 

31 

30 

.21644 

.22169 

4.5107 

.97630 

30 

30 

.23345 

.24008 

4.1653 

.97237 

30 

31 

672 

200 

.5045 

623 

29 

31 

373 

039 

.1600 

230 

29 

32 

701 

231 

.4983 

617 

28 

32 

401 

069 

.1547 

223 

28 

33 

729 

261 

.4922 

611 

27 

33 

429 

100 

.1493 

217 

27 

34 

758 

292 

.4860 

604 

26 

34 

458 

131 

.1441 

210 

26 

35 

.21786 

.22322 

4.4799 

.97598 

25 

35 

.23486 

.24162 

4.1388 

.97203 

25 

36 

814 

353 

.4737 

592 

24 

36 

514 

193 

.1335 

196 

24 

37 

843 

383 

.4676 

585 

23 

37 

542 

223 

.1282 

189 

23 

38 

871 

414 

.4615 

579 

22 

38 

671 

254 

.1230 

182 

22 

39 

899 

444 

.4555 

673 

21 

39 

699 

285 

.1178 

176 

21 

40 

.21928 

.22475 

4.4494 

.97566 

20 

40 

.23627 

.24316 

4.1126 

.97169 

20 

41 

956 

505 

.4434 

560 

19 

41 

656 

347 

.1074 

162 

19 

42 

.21985 

536 

.4373 

653 

18 

42 

684 

377 

.1022 

155 

18 

43 

.22013 

667 

.4313 

547 

17 

43 

712 

408 

.0970 

148 

17 

44 

041 

597 

.4253 

541 

16 

44 

740 

439 

.0918 

141 

16 

45 

.22070 

.22628 

4.4194 

.97534 

15 

45 

.23769 

.24470 

4.0867 

.97134 

15 

46 

098 

658 

.4134 

528 

14 

46 

797 

501 

.0815 

127 

14 

47 

126 

689 

.4075 

521 

13 

47 

825 

632 

.0764 

120 

13 

48 

155 

719 

.4015 

515 

12 

48 

853 

562 

.0713 

113 

12 

49 

183 

750 

.3956 

508 

11 

49 

882 

593 

.0662 

106 

11 

50 

.22212 

.22781 

4.3897 

.97502 

10 

50 

.23910 

.24624 

4.0611 

.97100 

10 

51 

240 

811 

.3838 

496 

9 

51 

938 

655 

.0560 

093 

9 

52 

268 

842 

.3779 

489 

8 

52 

966 

686 

.0509 

086 

8 

53 

297 

872 

.3721 

483 

7 

53 

.23995 

717 

.0459 

079 

7 

54 

325 

903 

.3662 

476 

6 

54 

.24023 

747 

.0408 

072 

6 

55 

.22353 

.22934 

4.3604 

.97470 

5 

55 

.24051 

.24778 

4.0358 

.97065 

6 

56 

382 

964 

.3546 

463 

4 

56 

079 

809 

.0308 

058 

4 

57 

410 

.22995 

.3488 

457 

3 

57 

108 

840 

.0257 

051 

3 

58 

438 

.23026 

.3430 

450 

2 

58 

136 

871 

.0207 

044 

2 

59 

467 

056 

.3372 

444 

1 

59 

164 

902 

.0158 

037 

1 

60 

.22495 

.23087 

4.3315 

.97437 

0 

60 

.24192 

.24933 

4.0108 

.97030 

0 

Cos 

Ctn 

Tan 

Sin 

1 

Cos 

Ctn   Tan 

Sin   f   1 

iq 

14°— Values  of  Trigononieti 

'ic  Functions  — 15° 

29 

i~ 

Sin 

Tan 

Ctn 

Cos 

1 

Sin 

Tan 

Ctn 

Cos 

0 

.24192 

.24933 

4.0108 

.97030 

60 

0 

.25882 

.26795 

3.7321 

.96593 

60 

1 

220 

964 

.0058 

023 

59 

1 

910 

826 

.7277 

J  585 

59 

2 

249 

.24995 

4.0009 

015 

58 

2 

938 

857 

.7234 

578 

58 

3 

277 

.25026 

3.9959 

008 

57 

3 

966 

888 

.7191 

570 

57 

4 

305 

056 

.9910 

.97001 

56 

4 

.25994 

920 

.7148 

562 

56 

5 

.24333 

.25087 

3.9861 

.96994 

55 

5 

.26022 

.26951 

3.7105 

.96555 

55 

6 

362 

118 

.9812 

987 

54 

6 

050 

.26982 

.7062 

547 

54 

7 

390 

149 

.9763 

980 

53 

7 

079 

27013 

.7019 

540 

53 

8 

418 

180 

.9714 

973 

52 

8 

107 

044 

.6976 

632 

62 

9 

446 

211 

.9665 

966 

51 

9 

135 

076 

.6933 

624 

61 

10 

.24474 

.25242 

3.9617 

.96959 

50 

10 

.26163 

.27107 

3.6891 

.96517 

50 

11 

503 

273 

.9568 

952 

49 

11 

191 

138 

.6848 

509 

49 

12 

531 

304 

.9520 

945 

48 

12 

219 

169 

.6806 

602 

48 

13 

559 

335 

.9471 

937 

47 

13 

247 

201 

.6764 

494 

47 

14 

587 

366 

.9423 

930 

46 

14 

275 

232 

.6722 

486 

46 

15 

.24615 

.25397 

3.9375 

.96923 

45 

15 

.26303 

.27263 

3.6680 

.96479 

45 

16 

644 

428 

.9327 

916 

44 

16 

331 

294 

.6638 

471 

44 

17 

672 

459 

.9279 

909 

43 

17 

359 

326 

.6596 

463 

43 

18 

700 

490 

.9232 

902 

42 

18 

387 

357 

.6554 

456 

42 

19 

728 

521 

.9184 

894 

41 

19 

415 

388 

.6512 

448 

41 

20 

.24756 

.25552 

3.9136 

.96887 

40 

20 

.26443 

.27419 

3.6470 

.96440 

40 

21 

784 

583 

.9089 

880 

39 

21 

471 

451 

.6429 

433 

39 

22 

813 

614 

.9042 

873 

38 

22 

500 

482 

.6387 

425 

38 

23 

841 

645 

.8995 

866 

37 

23 

528 

513 

.6346 

417 

37 

24 

869 

676 

.8947 

858 

36 

24 

556 

545 

.6305 

410 

36 

25 

.24897 

.25707 

3.8900 

.96851 

35 

25 

.26584 

.27576 

3.6264 

.96402 

35 

26 

925 

738 

.8854 

844 

34 

26 

612 

607 

.6222 

394 

34 

27 

954 

769 

.8807 

837 

33 

27 

640 

638 

.6181 

386 

33 

28 

.24982 

800 

.8760 

829 

32 

28 

668 

670 

.6140 

379 

32 

29 

.25010 

831 

.8714 

822 

31 

29 

696 

701 

.6100 

371 

31 

30 

.25038 

.25862 

3.8667 

.96815 

30 

30 

.26724 

.27732 

3.6059 

.96363 

30 

31 

066 

893 

.8621 

807 

29 

31 

752 

764 

.6018 

355 

29 

32 

094 

924 

.8575 

800 

28 

32 

780 

795 

.5978 

347 

28 

33 

122 

955 

.8528 

793 

27 

33 

808 

826 

.5937 

340 

27 

34 

151 

.25986 

.8482 

786 

26 

34 

836 

858 

.5897 

332 

2(i 

35 

.25179 

.26017 

3.8436 

.96778 

25 

35 

.26864 

.27889 

3.5856 

.96324 

25 

36 

207 

048 

.8391 

771 

24 

36 

892 

921 

.5816 

316 

24 

37 

235 

079 

.8345 

764 

23 

37 

920 

952 

.5776 

308 

23 

38 

263 

110 

.8299 

766 

22 

38 

948 

.27983 

.5736 

301 

22 

39 

291 

141 

.8254 

749 

21 

39 

.26976 

.28015 

.5696 

293 

21 

40 

.25320 

.26172 

3.8208 

.96742 

20 

40 

.27004 

.28046 

3.5656 

.96285 

20 

41 

348 

203 

.8163 

734 

19 

41 

032 

077 

.5616 

277 

19 

42 

376 

235 

.8118 

727 

18 

42 

060 

109 

.5576 

269 

18 

43 

404 

266 

.8073 

719 

17 

43 

088 

140 

.5536 

261 

17 

44 

432 

297 

.8028 

712 

16 

44 

116 

172 

.5497 

253 

1^ 

45 

.25460 

.26328 

3.7983 

.96705 

15 

45 

.27144 

.28203 

3.5457 

.96246 

15 

46 

488 

359 

.7938 

697 

14 

46 

172 

234 

.5418 

238 

14 

47 

516 

390 

.7893 

690 

13 

47 

200 

266 

.5379 

230 

13 

48 

545 

421 

.7848 

682 

12 

48 

228 

297 

.5339 

222 

12 

49 

573 

452 

.7804 

675 

11 

49 

256 

329 

.5300 

214 

11 

50 

.25601 

.26483 

3.7760 

.96667 

10 

50 

.27284 

.28360 

3.5261 

.96206 

10 

51 

629 

515 

.7715 

660 

9 

51 

312 

391 

.5222 

198 

9 

52 

657 

546 

.7671 

653 

8 

52 

340 

423 

.5183 

190 

8 

53 

685 

577 

.7627 

645 

7 

53 

368 

454 

.5144 

182 

7 

54 

713 

608 

.7583 

638 

6 

54 

396 

486 

.5105 

174 

6 

55 

.25741 

.26639 

3.7539 

.96630 

5 

55 

.27424 

.28517 

3.5067 

.96166 

5 

56 

769 

670 

.7495 

623 

4 

56 

452 

549 

.5028 

158 

4 

57 

798 

701 

.7451 

615 

3 

57 

480 

680 

.4989 

150 

3 

58 

826 

733 

.7408 

608 

2 

58 

508 

612 

.4951 

142 

^ 

59 

854 

764 

.7364 

600 

1 

59 

536 

643 

.4912 

134 

1 

60 

.25882 

.26795 

3.7321 

.9()593 

0 

60 

.27564 

.28675 

3.4874 

.96126 

0 

Gob 

Ctn 

Tan 

Sin 

/ 

Cos 

Ctn 

Tan 

Sin 

f 

30 

16°  — Values  of  Trigonometric  Functions  — 17" 

[" 

/ 

Sin 

Tan 

Ctn 

Cos 

f 

Sin 

Tan 

Ctn 

Cos 

0 

.27564 

.28675 

3.4874 

.96126 

60 

0 

.29237 

.30573 

3.2709 

.95630 

60 

1 

592 

706 

.4836 

118 

59 

1 

265 

605 

.2675 

622 

59 

2 

620 

738 

.4798 

110 

58 

2 

293 

637 

.2641 

613 

58 

3 

648 

769 

.4760 

102 

57 

3 

321 

669 

.2607 

605 

57 

4 

676 

801 

.4722 

094 

56 

4 

348 

700 

.2573 

596 

56 

5 

.27704 

.28832 

3.4684 

.96086 

55 

5 

.29376 

.30732 

3.2539 

.95588 

65 

6 

731 

864 

.4646 

078 

54 

6 

404 

764 

.250f3 

579 

54 

7 

759 

895 

.4608 

070 

53 

7 

432 

79(3 

.2472 

571 

53 

8 

787 

927 

.4570 

062 

52 

8 

460 

828 

.2438 

562 

52 

9 

815 

958 

.4533 

054 

51 

9 

487 

860 

.2405 

554 

51 

10 

.27843 

.28990 

3.4495 

.96046 

50 

10 

.29515 

.30891 

3.2371 

.95545 

60 

11 

871 

.29021 

.4458 

037 

49 

11 

543 

923 

.2338 

536 

49 

12 

899 

053 

.4420 

029 

48 

12 

571 

955 

.2305 

528 

48 

13 

927 

084 

.4383 

021 

47 

13 

599 

.30987 

.2272 

619 

47 

14 

955 

116 

.4346 

013 

46 

14 

626 

.31019 

.2238 

511 

46 

15 

.27983 

.29147 

3.4308 

.9(3005 

45 

15 

.29654 

.31051 

3.2205 

.95502 

45 

16 

.28011 

179 

.4271 

.95997 

44 

16 

682 

083 

.2172 

493 

44 

17 

039 

210 

.4234 

989 

43 

17 

710 

115 

.2139 

485 

43 

18 

067 

242 

.4197 

981 

42 

18 

737 

147 

.2106 

476 

42 

19 

095 

274 

.4160 

972 

41 

19 

765 

178 

.2073 

467 

41 

20 

.28123 

.29305 

3.4124 

.95964 

40 

20 

.29793 

.31210 

3.2041 

.95459 

40 

21 

150 

337 

.4087 

956 

39 

21 

821 

242 

.2008 

450 

39 

22 

178 

368 

.4050 

948 

38 

22 

849 

274 

.1975 

441 

38 

23 

206 

400 

.4014 

940 

37 

23 

876 

306 

.1943 

433 

37 

24 

234 

432 

.3977 

931 

36 

24 

904 

338 

.1910 

424 

36 

25 

.28262 

.29463 

3.3941 

.95923 

35 

25 

.29932 

.31370 

3.1878 

.95415 

36 

26 

290 

495 

.3904 

915 

34 

26 

960 

402 

.1845 

407 

34 

27 

318 

526 

.3868 

907 

33 

27 

.29987 

434 

.1813 

398 

33 

28 

346 

558 

.3832 

898 

32 

28 

.30015 

466 

.1780 

389 

32 

29 

374 

590 

.3796 

890 

31 

29 

043 

498 

.1748 

380 

31 

30 

.28402 

.29621 

3.3759 

.95882 

30 

30 

.30071 

.31530 

3.1716 

.95372 

30 

31 

429 

653 

.3723 

874 

29 

31 

098 

562 

.1684 

363 

29 

32 

457 

685 

.3687 

865 

28 

32 

126 

594 

.1652 

354 

28 

33 

485 

716 

.3652 

857 

27 

33 

154 

626 

.1620 

345 

27 

34 

513 

748 

.3616 

849 

26 

34 

182 

658 

.1588 

337 

26 

35 

.28541 

.29780 

3.3580 

.95841 

25 

35 

.30209 

.31690 

3.1556 

.95328 

25 

36 

569 

811 

.3544 

832 

24 

36 

237 

722 

.1524 

319 

24 

37 

597 

843 

.3509 

824 

23 

37 

265 

754 

.1492 

310 

23 

38 

625 

875 

.3473 

816 

22 

38 

'292 

786 

.1460 

301 

22 

39 

652 

906 

.3438 

807 

21 

39 

320 

818 

.1429 

293 

21 

40 

.28680 

.29938 

3.3402 

.95799 

20 

40 

.30348 

.31850 

3.1397 

.95284 

20 

41 

708 

.29970 

.3367 

791 

19 

41 

376 

882 

.1366 

275 

19 

42 

736 

.30001 

.3332 

782 

18 

42 

403 

914 

.1334 

266 

18 

43 

764 

033 

.3297 

774 

17 

43 

431 

946 

.1303 

257 

17 

M 

792 

065 

.3261 

76(3 

16 

44 

459 

.31978 

.1271 

248 

16 

45 

.28820 

.30097 

3.3226 

.95757 

15 

45 

.30486 

.32010 

3.1240 

.95240 

15 

46 

847 

128 

.3191 

749 

14 

46 

514 

042 

.1209 

231 

14 

47 

875 

160 

.3156 

740 

13 

47 

542 

074 

.1178 

222 

13 

48 

903 

192 

.3122 

732 

12 

48 

570 

106 

.1146 

213 

12 

49 

931 

224 

.3087 

724 

11 

49 

597 

139 

.1115 

204 

11 

50 

.28959 

.30255 

3.3052 

.95715 

10 

50 

.30625 

.32171 

3.1084 

.95195 

10 

51 

.28987 

287 

.3017 

707 

9 

51 

653 

203 

.1053 

186 

9 

52 

.29015 

319 

.2983 

698 

8 

52 

680 

235 

.1022 

177 

8 

53 

042 

351 

.2948 

690 

7 

53 

708 

267 

.0991 

168 

7 

54 

070 

382 

.2914 

681 

6 

54 

736 

299 

.0961 

159 

6 

55 

.29098 

.30414 

3.2879 

.95673 

5 

55 

.30763 

.32331 

3.0930 

.95150 

6 

56 

126 

446 

.2845 

664 

4 

56 

791 

363 

.0899 

142 

4 

57 

154 

478 

.2811 

656 

3 

57 

819 

396 

.0868 

133 

3 

58 

182 

509 

.2777 

647 

2 

58 

846 

428 

.0838 

124 

2 

59 

209 

541 

.2743 

639 

1 

59 

874 

460 

.0807 

115 

1 

60 

.29237 

.30573 

3.2709 

.95630 

0 

60 

.30902 

.32492 

3.0777 

.95106 

0 

Cos 

Ctn 

Tan 

Sin 

1 

Cos 

Ctn 

Tan 

Sin 

/ 

11] 


18°— Talues  of  Trigonometric  Functions  — 19° 


31 


/ 

Sin 

Tan 

Ctn 

Cos 

0 

.30902 

.32492 

3.0777 

.95106 

60 

1 

929 

524 

.0746 

097 

59 

2 

957 

556 

.0716 

088 

58 

3 

.30985 

588 

.0686 

079 

57 

4 

.31012 

621 

.0655 

070 

56 

5 

.31040 

.32653 

3.0625 

.95061 

55 

6 

068 

685 

.0595 

052 

54 

7 

095 

717 

.0565 

043 

53 

8 

123 

749 

.0535 

033 

52 

9 

151 

782 

.0505 

024 

51 

10 

.31178 

.32814 

3.0475 

.95015 

50 

11 

206 

846 

.0445 

.95006 

49 

12 

233 

878 

.0415 

.94997 

48 

13 

261 

911 

.0385 

988 

47 

14 

289 

943 

.0356 

979 

46 

15 

,31316 

.32975 

3.0326 

.94970 

45 

16 

344 

.33007 

.02% 

961 

44 

17 

372 

040 

.0267 

952 

43 

18 

399 

072 

.0237 

943 

42 

19 

427 

104 

.0208 

933 

41 

20 

.31454 

.33136 

3.0178 

.94924 

40 

21 

482 

169 

.0149 

915 

39 

22 

510 

201 

.0120 

906 

38 

23 

537 

233 

.0090 

897 

37 

24 

565 

266 

.0061 

888 

36 

25 

.31593 

.33298 

3.0032 

.94878 

35 

26 

620 

330 

3.0003 

869 

34 

27 

648 

363 

2.9974 

860 

33 

28 

675 

395 

.9945 

851 

32 

29 

703 

427 

.9916 

842 

31 

30 

.31730 

.33460 

2.9887 

.94832 

30 

31 

758 

492 

.9858 

823 

29 

32 

786 

524 

.9829 

814 

28 

33 

813 

557 

.9800 

805 

27 

34 

841 

589 

.9772 

795 

26 

35 

.31868 

.33621 

2.9743 

Mim 

25 

36 

896 

654 

.9714 

Til 

24 

37 

923 

686 

.9686 

768 

23 

38 

951 

718 

.9657 

758 

22 

39 

.31979 

751 

.9629 

749 

21 

40 

.32006 

.33783 

2.9600 

.94740 

20 

41 

034 

816 

.9572 

730 

19 

42 

061 

848 

.9544 

721 

18 

43 

089 

881 

.9515 

712 

17 

44 

116 

913 

.9487 

792 

16 

45 

.32144 

.33945 

2.9459 

.94693 

15 

46 

171 

.33978 

.9431 

684 

14 

47 

199 

.34010 

.9403 

674 

13 

48 

227 

043 

.9375 

665 

12 

49 

254 

075 

.9347 

656 

11 

50 

.32282 

.34108 

2.9319 

.94(346 

10 

51 

309 

140 

.9291 

637 

9 

52 

337 

173 

.9263 

627 

8 

53 

364 

205 

.9235 

618 

7 

54 

392 

238 

.9208 

609 

6 

55 

.32419 

.34270 

2.9180 

.94599 

5 

56 

447 

303 

.9152 

590 

4 

57, 

474 

335 

.9125 

580 

3 

58" 

502 

368 

.9097 

571 

2 

59 

529 

400 

.9070 

561 

1 

60 

.32557 

.34433 

2.9042 

.94552 

0 

Cos 

Ctn 

Tan 

Sin 

/ 

1 

Sin 

Tan 

Ctn 

Cos 

0 

.32557 

.34433 

2.9042 

.94552 

60 

1 

584 

465 

.9015 

642 

59 

2 

612 

498 

.8987 

533 

58 

3 

639 

530 

.8960 

523 

57 

4 

667 

563 

.8933 

514 

56 

5 

.32694 

.3459(3 

2.8^X)5 

.94504 

55 

6 

722 

628 

.8878 

495 

54 

7 

749 

661 

.8851 . 

485 

53 

8 

777 

693 

.8824 

476 

52 

9 

804 

726 

.8797 

466 

51 

10 

.32832 

.34758 

2.8770 

.94457 

50 

11 

859 

791 

.8743 

447 

49 

12 

887 

824 

.8716 

438 

48 

13 

'914 

856 

.8689 

428 

47 

14 

942 

889 

.8662 

418 

46 

15 

.32969 

.34922 

2.8636 

.94409 

45 

16 

.32997 

954 

.8(309 

399 

44 

17 

.33024 

.31987 

.8582 

390 

43 

18 

051 

.35020 

.8556 

380 

42 

19 

079 

052 

.8529 

370 

41 

20 

.33106 

.35085 

2.8502 

.94361 

40 

21 

134 

118 

.8476 

351 

39 

22 

161 

150 

.8449 

342 

38 

23 

189 

183 

.8423 

332 

37 

24 

216 

216 

.8397 

322 

36 

25 

.33244 

.35248 

2.8370 

.94313 

35 

26 

271 

281 

.8344 

303 

34 

27 

298 

314 

.8318 

293 

33 

28 

326 

346 

.8291 

284 

32 

29 

353 

379 

.8265 

274 

31 

30 

.33381 

.35412 

2.8239 

.94264 

30 

31 

408 

445 

.8213 

254 

29 

32 

436 

477 

.8187 

•245 

28 

33 

463 

510 

.8161 

235 

27 

34 

m) 

543 

.8135 

225 

26 

35 

.33518 

.35576 

2.8109 

.94215 

25 

36 

545 

608 

.8083 

206 

24 

37 

573 

641 

.8057 

196 

23 

38 

600 

674 

.8032 

186 

22 

39 

627 

707 

.8006 

176 

21 

40 

.33(355 

.35740 

2.7980 

.94167 

20 

41 

682 

772 

.7955 

157 

19 

42 

710 

805 

.7929 

147 

18 

43 

737 

838 

.7903 

137 

17 

44 

7(>4 

871 

.7878 

127 

16 

45 

.33792 

.35904 

2.7852 

.94118 

15 

46 

819 

937 

.7827 

108 

14 

47 

846 

.35969 

.7801 

098 

13 

48 

874 

.36002 

.7776 

088 

12 

49 

901 

035 

.7751 

078 

11 

50 

.33929 

.36068 

2.7725 

.94068 

10 

51 

956 

101 

.7700 

058 

9 

52 

.33983 

134 

.7675 

049 

8 

53 

.34011 

167 

.7650 

039 

7 

54 

038 

199 

.7625 

029 

6 

55 

.34065 

.36232 

2.7600 

.94019 

5 

56 

093 

265 

.7575 

.94009 

4 

57 

120 

298 

.7550 

.93999 

3 

58 

147 

331 

.7525 

989 

2 

59 

175 

364 

.7500 

979 

1 

60 

.34202 

.36397 

2.7475 

.93969 

0 

Cos 

Ctn 

Tan 

Sin 

/ 

wto 


nap 


32         20°  —  Values  of  Trigonometric  Functions  —  21* 


/ 

Sin 

Tan 

Ctn 

Cos 

0 

.34202 

.36397 

2.7475 

.93969 

60 

1 

229 

430 

.7450 

959 

69 

2 

257 

463 

.7425 

949 

58 

3 

284 

496 

.7400 

939 

57 

4 

311 

529 

.7376 

929 

66 

5 

.34339 

.36562 

2.7351 

.93919 

55 

6 

366 

595 

■  7326 

909 

54 

7 

393 

628 

.7302 

899 

53 

8 

421 

661 

.7277 

889 

52 

9 

448 

694 

.7253 

879 

51 

10 

.34476 

.36727 

2.7228 

.93869 

50 

11 

603 

760 

.7204 

859 

49 

12 

630 

793 

.7179 

849 

48 

13 

657 

826 

.7155 

839 

47 

14 

684 

859 

.7130 

829 

46 

15 

.34612 

.36892 

2.7m) 

.93819 

45 

16 

639 

925 

.7082 

809 

44 

17 

666 

958 

.7058 

799 

43 

18 

694 

.36991 

.7034 

789 

42 

19 

721 

.37024 

.7009 

779 

41 

20 

.34748 

.37057 

2.6985 

.93769 

40 

21 

775 

090 

.6961 

759 

39 

22 

803 

123 

.6937 

748 

38 

23 

830 

157 

.6913 

738 

37 

24 

857 

190 

.6889 

728 

36 

25 

.34884 

.37223 

2.6865 

.93718 

35 

26 

912 

266 

.6841 

708 

34 

27 

939 

289 

.6818 

698 

33 

28 

966 

322 

.6794 

688 

32 

29 

.34993 

355 

.6770 

677 

31 

30 

.35021 

.37388 

2.6746 

.93667 

30 

31 

048 

422 

.6723 

657 

29 

32 

676 

455 

.6699 

647 

28 

33 

102 

488 

.6675 

637 

27 

34 

130 

521 

.6652 

626 

26 

35 

•  .35157 

.37554 

2.6628 

.93616 

25 

36 

184 

588 

.6605 

606 

24 

37 

211 

621 

.6581 

596 

23 

38 

239 

654 

.6558 

585 

22 

39 

266 

687 

.65:34 

575 

21 

40 

.35293 

.37720 

2.6511 

.93565 

20 

41 

320 

754 

.6488 

655 

19 

42 

347 

787 

.6464 

644 

18 

43 

375 

820 

.6441 

534 

17 

44 

402 

853 

.6418 

524 

16 

45 

.35429 

.37887 

2.6395 

.93514 

15 

46 

456 

920 

.6371 

503 

14 

47 

484 

953 

.6:348 

493 

13 

48 

511 

.37986 

.6325 

483 

12 

49 

538 

.38020 

.6302 

472 

11 

50 

.35565 

.38053 

2.6279 

.93462 

10 

61 

592 

086 

.6256 

452 

9 

62 

619 

120 

.6233 

441 

8 

63 

647 

153 

.6210 

431 

7 

54 

674 

186 

.6187 

420 

6 

55 

.35701 

.38220 

2.6165 

.93410 

5 

56 

728 

253 

.6142 

400 

4 

57 

756 

286 

.6119 

389 

3 

58 

782 

320 

.6096 

379 

2 

59 

810 

353 

.6074 

368 

1 

60 

.36837 

.38386 

2  6051 

.93358 

0 

Cos 

Ctn 

Tan 

Sin 

/ 

1 

Sin 

Tan 

Ctn 

Cos 

~~0 

.35837 

.38386 

2.6051 

.93358 

60 

1 

864 

420 

.6028 

348 

59 

2 

891 

453 

.6006 

337 

58 

3 

918 

487 

.6983 

327 

67 

4 

945 

520 

.5961 

316 

56 

5 

.35973 

.38553 

2.6938 

.93306 

55 

6 

.36000 

687 

.5916 

295 

64 

7 

027 

620 

.5893 

285 

53 

8 

•  054 

664 

.6871 

274 

52 

9 

081 

687 

.5848 

264 

51 

10 

.36108 

.38721 

2.5826 

.93253 

50 

11 

135 

754 

.5804 

243 

49 

12 

162 

787 

.5782 

232 

48 

13 

l^X) 

821 

.5759 

222 

47 

14 

217 

854 

.5737 

211 

46 

15 

.36244 

.38888 

2.5715 

.93201 

45 

16 

271 

921 

.5693 

190 

44 

17 

298 

955 

.5671 

180 

43 

18 

325 

.38988 

.5649 

169 

42 

19 

352 

.39022 

.5627 

159 

41 

20 

.36379 

.39055 

2.5605 

.93148 

40 

21 

406 

089 

.5583 

137 

39 

22 

434 

122 

.5561 

127 

38 

23 

461 

156 

.5539 

116 

37 

24 

488 

190 

.6617 

106 

36 

25 

.36515 

.39223 

2.5495 

.93095 

35 

26 

542 

257 

.6473 

084 

34 

27 

669 

290 

.6452 

074 

33 

28 

696 

324 

.5430 

063 

32 

29 

623 

357 

.5408 

052 

31 

30 

.36650 

.39391 

2.5386 

.93042 

30 

31 

677 

425 

.5365 

031 

29 

32 

704 

458 

.5343 

020 

28 

33 

731 

492 

.5322 

.93010 

27 

34 

758 

526 

.5300 

.92999 

26 

35 

.36785 

.39559 

2.5279 

.92988 

25 

36 

812 

693 

.5257 

978 

24 

37 

839 

626 

.5236 

mi 

23 

38 

867 

660 

.5214 

956 

22 

39 

894 

694 

.6193 

945 

21 

40 

.36921 

.39727 

2.5172 

.92935 

20 

41 

948 

761 

.5160 

924 

19 

42 

.36975 

796 

.5129 

913 

18 

43 

.37002 

829 

.5108 

902 

17 

44 

029 

862 

.5086 

892 

16 

45 

.37056 

.39896 

2.5065 

.92881 

15 

46 

083 

930 

.5044 

870 

14 

47 

110 

963 

.5023 

859 

13 

48 

137 

.39997 

.5002 

849 

12 

49 

164 

.40031 

.4981 

838 

11 

50 

.37191 

.40065 

2.4960 

.92827 

10 

51 

218 

098 

.4939 

816 

9 

62 

245 

132 

.4918 

805 

8 

53 

272 

166 

.4897 

794 

7 

64 

299 

200 

.4876 

784 

6 

55 

.37326 

.40234 

2.4865 

.92773 

5 

56 

353 

267 

.4834 

762 

4 

57 

380 

301 

.4813 

751 

3 

58 

407 

335 

.4792 

740 

2 

69 

434 

369 

.4772 

729 

1 

60 

.37461 

.40403 

2.4751 

.92718 

0 

Cos 

Ctn 

Tan 

Sin 

1 

aQ9 


11] 

2 

2°— Values  of  Trigonometric  Functions  —  23° 

33 

/ 

Sin 

Tan 

Ctn 

Cos 

f 

Sin 

1  Tan 

Ctn 

Cos 

0 

.37461 

.40403 

2.4751 

.92718 

60 

0 

.39073 

.42447 

2.3559 

.92050 

60 

1 

488 

436 

.4730 

707 

59 

1 

100 

482 

.3539 

039 

59 

2 

515 

470 

.4709 

697 

58 

2 

127 

616 

.3520 

028 

58 

3 

542 

504 

.4689 

686 

57 

3 

153 

551 

.3501 

016 

57 

4 

569 

538 

.4668 

675 

56 

4 

180 

585 

.3483 

.92005 

56 

5 

.37595 

.40572 

2.4648 

.92664 

55 

5 

.39207 

.42619 

2.3464 

.91994 

55 

6 

622 

606 

.4627 

653 

54 

6 

234 

(354 

.3445 

982 

54 

7 

649 

640 

.4606 

642 

53 

7 

260 

688 

.^426 

971 

53 

8 

676 

674 

.4586 

631 

52 

8 

287 

722 

.3407 

959 

52 

9 

703 

707 

.4566 

620 

51 

9 

314 

757 

.3388 

948 

51 

10 

,37730 

.40741 

2.4545 

.92609 

50 

10 

.39341 

.42791 

2.3369 

.91936 

50 

11 

757 

775 

/.4525 

598 

49 

11 

3()7 

826 

.3351 

925 

49 

12 

784 

809 

.4504 

587 

48 

12 

394 

860 

.3332 

914 

48 

13 

811 

843 

.4484 

576 

47 

13 

421 

894 

.3313 

902 

47 

14 

838 

877 

.4464 

565 

46 

14 

448 

929 

.3294 

891 

46 

15 

.37865 

.40911 

2.4443 

.92554 

45 

15 

.39474 

.42963 

2.3276 

.91879 

45 

16 

892 

945 

.4423 

543 

44 

16 

501 

.42998 

.3257 

868 

44 

17 

919 

.40979 

.4403 

532 

43 

17 

528 

.43032 

.3238 

856 

43 

18 

946 

.41013 

.4383 

521 

42 

18 

555 

067 

.3220 

845 

42 

19 

973 

047 

.4362 

510 

41 

19 

581 

101 

.3201 

833 

41 

20 

.37999 

.41081 

2.4342 

.92499 

40 

20 

.39608 

.43136 

2.3183 

.91822 

40 

21 

.38026 

115 

.4322 

488 

39 

21 

635 

170 

.3164 

810 

39 

22 

053 

149 

.4302 

477 

38 

22 

661 

205 

.3146 

799 

38 

23 

080 

183 

.•4282 

466 

37 

23 

688 

239 

.3127 

787 

37 

24 

107 

217 

.4262 

455 

36 

24 

715 

274 

.3109 

775 

36 

25 

.38134 

.41251 

2.4242 

.92444 

35 

25 

.39741 

.43308 

2.3090 

.91764 

35 

26 

161 

285 

.4222 

432 

34 

26 

768 

343 

.3072 

752 

34 

27 

188 

319 

!4202 

421 

33 

27 

795 

378 

.3053 

741 

33 

28 

215 

353 

.4182 

410 

32 

28 

822 

412 

.3035 

729 

32 

29 

241 

387 

.4162 

399 

31 

29 

848 

447 

.3017 

718 

31 

30 

.38268 

.41421 

2.4142 

.92.388 

30 

30 

.39875 

.43481 

2.2998 

.91706 

30 

31 

295 

455 

.4122 

377 

29 

31 

902 

516 

.2980 

694 

29 

32 

322 

490 

.4102 

366 

28 

32 

928 

550 

.2962 

683 

28 

33 

349 

524 

.4083 

355 

27 

33 

955 

685 

.2944 

671 

27 

34 

376 

558 

.4063 

343 

26 

34 

.39982 

620 

.2925 

660 

26 

35 

.38403 

.41592 

2.4043 

.92332 

25 

35 

.40008 

.43654 

2.2907 

.91648 

25 

36 

430 

626 

.4023 

321 

24 

36 

035 

689 

.2889 

636 

24 

37 

456 

660 

.4004 

310 

23 

37 

062 

724 

.2871 

625 

23 

38 

483 

694 

.3984 

299 

22 

38 

088 

758 

.2853 

613 

22 

39 

510 

728 

.3964 

287 

21 

39 

115 

793 

.2835 

601 

21 

40 

.38537 

.41763 

2.3945 

.92276 

20 

40 

.40141 

.43828 

2.2817 

.91590 

20 

41 

564 

797 

.3925 

265 

19 

41 

168 

862 

.2799 

578 

19 

42 

591 

831 

.390\'5 

254 

18 

42 

195 

897 

.2781 

566 

18 

43 

617 

865 

.3886 

243 

17 

43 

221 

932 

.2763 

555 

17 

44 

644 

899 

.3867 

231 

16 

44 

248 

.43<)66 

.2745 

543 

\^ 

45 

.38671 

.41933 

2.3847 

.92220 

15 

45 

.40275 

.44001 

2.2727 

.91531 

15 

46 

698 

.419()8 

.3828 

209 

14 

46 

301 

036 

.2709 

519 

14 

47 

725 

.42002 

.3808 

198 

13 

47 

328 

071 

.2691 

508 

13 

48 

752 

036 

.3789 

186 

12 

48 

355 

105 

.2673 

496 

12 

49 

778 

070 

.3770 

175 

11 

49 

381 

140 

.2655 

484 

11 

50 

.38805 

.42105 

2.3750 

.92164 

10 

50 

.40408 

.44175 

2.2637 

.91472 

10 

51 

832 

139 

.3731 

152 

9 

51 

434 

210 

.2620 

461 

9 

52 

859 

173 

.3712 

141 

8 

52 

461 

244 

.2602 

449 

8 

53 

886 

207 

.3693 

130 

7 

53 

488 

279 

.2584 

437 

7 

54 

912 

242 

.3673 

119 

6 

54 

514 

314 

.2566 

425 

6 

55 

.38939 

.42276 

2.3654 

.92107 

5 

55 

.40541 

.44349 

2.2549 

.91414 

5 

56 

966 

310 

.3635 

096 

4 

56 

567 

384 

.2531 

402 

4 

57 

.38993 

345 

.3616 

085 

3 

57 

594 

418 

.2513 

390 

3 

58 

.39020 

379 

.3597 

073 

2 

58 

621 

453 

.2496 

378 

2 

59 

046 

413 

.3578 

062 

1 

59 

647 

488 

.2478 

366 

1 

60 

.39073 

.42447 

2.3559 

.92050 

0 

60 

.40674 

.44523 

2.2460 

.91355 

0 

Cos 

Gtn 

Tan 

Sin 

/ 

Cos 

Ctn 

Tan 

Sin 

1 

34 

2r  — Values  of  Trigonometric  Functions -^  25^ 

Pl 

/ 

Sin 

Tan 

Ctn 

Cos 

1 

Sin 

Tan 

Ctn 

Cos 

0 

.40674 

.44523 

2.2460 

.91355 

60 

0 

.42262 

.46631 

2.1445 

.90631 

60 

1 

700 

558 

.2443 

343 

69 

1 

288 

666 

.1429 

618 

59 

2 

727 

593 

.2425 

331 

68 

2 

316 

702 

.1413 

606 

58 

3 

753 

627 

.2408 

319 

67 

3 

341 

737 

.1396 

594 

67 

4 

780 

662 

.2390 

307 

56 

4 

367 

772 

.1380 

582 

56 

5 

.40806 

.44697 

2.2373 

.91295 

55 

5 

.42394 

.46808 

2.1364 

.90569 

55 

6 

833 

732 

.2356 

283 

54 

6 

420 

843 

.1348 

557 

54 

7 

860 

767 

.2338 

272 

53 

7 

446 

879 

.1332 

645 

53 

8 

886 

802 

.2320 

260 

62 

8 

473 

914 

.1316 

532 

52 

9 

913 

837 

.2303 

248 

61 

9 

499 

950 

.1299 

620 

51 

10 

.40939 

.44872 

2.2286 

.91236 

50 

10 

.42626 

.46986 

2.1283 

.90507 

60 

11 

966 

907 

.2268 

224 

49 

11 

652 

.47021 

.1267 

495 

49 

12 

.40992 

942 

.2251 

212 

48 

12 

678 

056 

.1251 

483 

48 

13 

.41019 

.44977 

.2234 

200 

47 

13 

604 

092 

.1235 

470 

47 

14 

045 

.45012 

.2216 

188 

46 

14 

631 

128 

.1219 

458 

46 

15 

.41072 

.45047 

2.2199 

.91176 

45 

15 

.42657 

.47163 

2.1203 

.90446 

45 

16 

098 

082 

.2182 

164 

44 

16 

683 

199 

.1187 

433 

44 

17 

125 

117 

.2165 

152 

43 

17 

709 

234 

.1171 

421 

43 

18 

151 

152 

.2148 

140 

42 

18 

736 

270 

.1155 

408 

42 

19 

178 

187 

.2130 

128 

41 

19 

762 

305 

.1139 

396 

41 

20 

.41204 

.45222 

2.2113 

.91116 

40 

20 

.42788 

.47341 

2.1123 

.90383 

40 

21 

231 

257 

.2096 

104 

39 

21 

815 

377 

.1107 

371 

39 

22 

257 

292 

.2079 

092 

38 

22 

841 

412 

.1092 

358 

38 

23 

284 

327 

.2062 

080 

37 

23 

867 

448 

.1076 

346 

37 

24 

310 

362 

.2046 

068 

36 

24 

894 

483 

.1060 

334 

36 

25 

.41337 

.45397 

2.2028 

.91056 

35 

25 

.42920 

.47519 

2.1044 

.90321 

35 

26 

363 

432 

.2011 

044 

34 

26 

946 

555 

.1028 

309 

34 

27 

390 

467 

.1994 

032 

33 

27 

972 

590 

.1013 

296 

33 

28 

416 

502 

.1977 

020 

32 

28 

.42999 

626 

.0997 

284 

32 

29 

443 

538 

.1960 

.91008 

31 

29 

.43025 

662 

.0981 

271 

31 

30 

.41469 

.46573 

2.1943 

.90996 

30 

30 

.43051 

.47698 

2.0965 

.90259 

30 

31 

496 

608 

.1926 

984 

29 

31 

077 

733 

.0950 

246 

29 

32 

522 

643 

.1909 

972 

28 

32 

104 

769 

.0934 

233 

28 

33 

549 

678 

.1892 

960 

27 

33 

130 

805 

.0918 

221 

27 

34 

675 

713 

.1876 

948 

26 

34 

156 

840 

.0903 

208 

26 

35 

.41602 

.45748 

2.1859 

.90936 

25 

35 

.43182 

.47876 

2.0887 

.90196 

25 

36 

628 

784 

.1842 

924 

24 

36 

209 

912 

.0872 

183 

24 

37 

655 

819 

.1825 

911 

23 

37 

236 

948 

.0866 

171 

23 

38 

681 

864 

.1808 

899 

22 

38 

261 

.47984 

.0840 

158 

22 

39 

707 

889 

.1792 

887 

21 

39 

287 

.48019 

.0826 

146 

21 

40 

.41734 

.46924 

2.1775 

.90875 

20 

40 

.43313 

.48055 

2.0809 

.90133 

20 

41 

760 

960 

.1758 

863 

19 

41 

340 

091 

.0794 

120 

19 

42 

787 

.46995 

.1742 

851 

18 

42 

366 

127 

.0778 

108 

18 

43 

813 

.460130 

.1725 

839 

17 

43 

392 

163 

.0763 

095 

17 

44 

840 

065 

.1708 

826 

16 

44 

418 

198 

.0748 

082 

16 

45 

.41866 

.46101 

2.1692 

.90814 

15 

45 

.43445 

.48234 

2.0732 

.90070 

15 

46 

892 

136 

.1675 

802 

14 

46 

471 

270 

.0717 

057 

14 

47 

919 

171 

.1659 

790 

13 

47 

497 

306 

.0701 

045 

13 

48 

945 

206 

.1642 

778 

12 

48 

623 

342 

.0686 

032 

12 

49 

972 

242 

.1625 

766 

11 

49 

649 

378 

.0671 

019 

11 

50 

.41998 

.46277 

2.1609 

.90753 

10 

50 

.43575 

.48414 

2.0655 

.90007 

10 

51 

.42024 

312 

.1592 

741 

9 

61 

602 

450 

.0640 

.89994 

9 

62 

051 

348 

.1576 

729 

8 

52 

628 

486 

.0625 

981 

8 

53 

077 

383 

.1560 

717 

7 

63 

654 

621 

.0609 

968 

7 

54 

104 

418 

.1543 

704 

6 

64 

680 

567 

.0594 

956 

6 

55 

.42130 

.46454 

2.1527 

.90692 

5 

55 

.43706 

.48593 

2.0679 

.89943 

5 

56 

166 

489 

.1510 

680 

4 

56 

733 

629 

.0564 

930 

4 

57 

183 

525 

.1494 

668 

3 

57 

759 

665 

.0549 

918 

3 

58 

209 

560 

.1478 

665 

2 

68 

785 

701 

.0633 

905 

2 

59 

235 

595 

.1461 

643 

1 

59 

811 

737 

.0518 

892 

1 

60 

.42262 

.46631 

2.1445 

.90631 

0 

60 

.43837 

.48773 

2  0503 

.89879 

0 

Cos 

Ctn 

Tan 

Sin 

1 

Cos 

Ctn 

Tan 

Sin 

; 

26° — Values  of  Trigonometric  Functions  —  27° 


35 


/ 

Sin 

Tan 

Ctn 

Cos 

/ 

Sin 

Tan 

Ctn 

Cos 

0 

.43837 

.48773 

2.0503 

.89879 

60 

0 

.45399 

.50953 

1.9626 

.89101 

60 

1 

863 

809 

.0^88 

867 

59 

1 

425 

.50989 

.9612 

087 

59 

2 

889 

845 

.0473 

854 

58 

2 

451 

.51026 

.9598 

074 

58 

3 

916 

881 

.0458 

841 

57 

3 

477 

063 

.9584 

061 

57 

4 

942 

917 

.0443 

828 

56 

4 

503 

099 

.9570 

048 

56 

5 

.43968 

.48953 

2.0428 

.89816 

55 

5 

.45529 

.51136 

1.9556 

.89035 

55 

6 

.43994 

.48989 

.0413 

803 

54 

6 

554 

173 

.9542 

021 

54 

7 

.44020 

.49026 

.0398 

790 

53 

7 

580 

209 

.9528 

.89008 

53 

8 

046 

062 

.0383 

777 

52 

8 

606 

246 

.9514 

.88995 

52 

9 

072 

098 

.0368 

764 

51 

9 

632 

283 

.9500 

981 

51 

10 

.44098 

.49134 

2.0353 

.89752 

50 

10 

.45658 

.51319 

1.9486 

.88968 

50 

11 

124 

170 

.0338 

739 

4<) 

11 

684 

356 

.9472 

955 

49 

12 

151 

206 

.0323 

726 

48 

12 

710 

393 

.9458 

942 

48 

13 

177 

242 

.0308 

713 

47 

13 

736 

430 

.9444 

928 

47 

14 

203 

278 

.0293 

700 

46 

14 

762 

467 

.9430 

915 

46 

15 

.44229 

.49315 

2.0278 

.89687 

45 

15 

.45787 

.51503 

1.9416 

.88902 

45 

16 

255 

351 

.0263 

674 

44 

16 

813 

540 

.9402 

888 

44 

17 

281 

387 

.0248 

662 

43 

17 

839 

577 

.9388 

875 

43 

18 

307 

423 

.0233 

649 

42 

18 

865 

614 

.9375 

862 

42 

19 

333 

459 

.0219 

636 

41 

19 

891 

651 

.9361 

848 

41 

20 

.44359 

.49495 

2.0204 

.89623 

40 

20 

.45917 

.51688 

1.9347 

.88835 

40 

21 

385 

532 

.0189 

610 

39 

21 

942 

724 

.9333 

822 

39 

22 

411 

568 

.0174 

597 

38 

22 

968 

761 

.9319 

808 

38 

23 

437 

604 

.0160 

584 

37 

23 

.45994 

798 

.9306 

795 

37 

24 

464 

640 

.0145 

571 

36 

24 

.46020 

835 

.9292 

782 

36 

25 

.44490 

.49677 

2.0130 

.89558 

35 

25 

.46046 

.51872 

1.9278 

.88768 

85 

26 

516 

713 

.0115 

545 

34 

26 

072 

909 

.9265 

755 

34 

27 

542 

749 

.0101 

532 

33 

27 

097 

946 

.9251 

741 

33 

28 

568 

786 

.0086 

519 

32 

28 

123 

.51983 

.9237 

728 

32 

29 

594 

822 

.0072 

506 

31 

29 

149 

.52020 

.9223 

715 

31 

30 

.44620 

.49858 

2.0057 

.89493 

30 

30 

.46175 

.52057 

1.9210 

.88701 

30 

31 

646 

894 

.0042 

480 

29 

31 

201 

094 

.9196 

688 

29 

32 

672 

931 

.0028 

467 

28 

32 

226 

131 

.9183 

674 

28 

33 

698 

.49967 

2.0013 

454 

27 

33 

252 

168 

.9169 

661 

27 

34 

724 

.50004 

1.9999 

441 

26 

34 

278 

205 

.9155 

647 

26 

35 

.44750 

.50040 

1.9984 

.89428 

25 

35 

.46304 

.52242 

1.9142 

.88634 

25 

36 

776 

076 

.9970 

415 

24 

36 

330 

279 

.9128 

620 

24 

37 

802 

113 

.9955 

402 

23 

37 

355 

316 

.9115 

607 

23 

38 

828 

149 

.9941 

389 

22 

38 

381 

353 

.9101 

693 

22 

39 

854 

185 

.9926 

376 

21 

39 

407 

390 

.9088 

580 

21 

40 

.44880 

.50222 

1.9912 

.89363 

20 

40 

.46433 

.52427 

1.9074 

.88566 

20 

41 

906 

258 

.9897 

350 

19 

41 

458 

464 

.9061 

553 

19 

42 

932 

295 

.9883 

337 

18 

42 

484 

501 

.9047 

539 

18 

43 

958 

331 

.9868 

324 

17 

43 

510 

538 

.9034 

526 

17 

44 

.44984 

368 

.9854 

311 

16 

44 

536 

575 

.9020 

612 

16 

45 

.45010 

.50404 

1.9840 

.89298 

15 

45 

.46561 

.52613 

1.9007 

.88499 

15 

46 

036 

441 

.9825 

285 

14 

46 

587 

650 

.8993 

485 

14 

47 

062 

477 

.9811 

272 

13 

47 

613 

687 

.8980 

472 

13 

48 

088 

514 

.9797 

259 

12 

48 

639 

724 

.8967 

458 

12 

49 

114 

550 

.9782 

245 

11 

49 

664 

761 

.8953 

445 

11 

50 

.45140 

.50587 

1.9768 

.89232 

10 

50 

.46690 

.52798 

1.8940 

.88431 

10 

51 

166 

623 

.9754 

219 

9 

51 

716 

836 

.8927 

417 

9 

52 

192 

660 

.9740 

206 

8 

52 

742 

873 

.8913 

404 

8 

53 

218 

696 

.9725 

193 

7 

53 

767 

910 

.8900 

390 

7 

54 

243 

733 

.9711 

180 

6 

54 

793 

947 

.8887 

377 

6 

55 

.45269 

.50769 

1.9697 

.89167 

5 

55 

.46819 

.52985 

1.8873 

.88363 

5 

56 

295 

806 

.9683 

153 

4 

56 

844 

.53022 

.8860 

349 

4 

•'57 

321 

843 

.9669 

140 

3 

57 

870 

059 

.8847 

336 

3 

58 

347 

879 

.9654 

127 

2 

58 

896 

096 

.8834 

322 

2 

59 

373 

916 

.9640 

114 

1 

59 

921 

134 

.8820 

308 

1 

60 

.45399 

.50953 

1.9626 

.89101 

0 

60 

.46947 

.53171 

1.8807 

.88295 

0 

Cos 

Ctn 

Tan 

Sin 

/ 

Cos 

Ctn 

Tan 

Sin 

/ 

A0»o 


36 

28°  — Values  of  Trigonometric  Functions  —  29° 

m 

r 

Sin 

Tan 

Ctn 

Cos 

/ 

Sin 

Tan 

Ctn 

Cos 

0 

.46947 

.53171 

1.8807 

.88295 

60 

0 

.48481 

.55431 

1.8040 

.87462 

60 

1 

973 

208 

.8794 

281 

59 

1 

506 

469 

.8028 

448 

59 

2 

.46999 

246 

.8781 

267 

58 

2 

532 

507 

.8016 

434 

58 

3 

.47024 

283 

.8768 

254 

57 

3 

557 

545 

.8003 

420 

57 

4 

050 

320 

.8755 

240 

56 

4 

583 

583 

.7991 

406 

56 

5 

.47076 

.53358 

1.8741 

.88226 

55 

5 

.48608 

.55621 

1.7979 

.87391 

55 

6 

101 

395 

.8728 

213 

54 

6 

634 

659 

.7966 

377 

54 

7 

127 

432 

.8715 

199 

53 

7 

659 

697 

.7954 

363 

53 

8 

153 

470 

.8702 

185 

52 

8 

684 

736 

.7942 

349 

52 

9 

178 

507 

.8689 

172 

51 

9 

710 

774 

.7930 

335 

51 

10 

.47204 

.53545 

1.8676 

.88158 

50 

10 

.48735 

.55812 

1.7917 

.87321 

50 

11 

229 

582 

.8663 

144 

49 

11 

761 

850 

.7905 

306 

49 

12 

255 

620 

.8650 

130 

48 

12 

786 

888 

.7893 

292 

48 

13 

281 

657 

.8637 

117 

47 

13 

811 

926 

.7881 

278 

47 

14 

306 

694 

.8624 

103 

46 

14 

837 

.55964 

.7868 

264 

46 

15 

.47332 

.53732 

1.8611 

.88089 

45 

16 

.48862 

.56003 

1.7856 

.87250 

45 

16 

358 

769 

.8598 

075 

44 

16 

888 

041 

.7844 

235 

44 

17 

383 

807 

.8585 

062 

43 

17 

913 

079 

.7832 

221 

43 

18 

409 

844 

.8572 

048 

42 

18 

938 

117 

.7820 

207 

42 

19 

434 

882 

.8559 

034 

41 

19 

964 

156 

.7808 

193 

41 

20 

.47460 

.53920 

1.8546 

.88020 

40 

20 

.48989 

.56194 

1.7796 

.87178 

40 

21 

486 

957 

.8533 

.88006 

39 

21 

.49014 

232 

.7783 

164 

39 

22 

511 

.53995 

.8520 

.87993 

38 

22 

040 

270 

.7771 

150 

38 

23 

537 

.54032 

.8507 

979 

37 

23 

065 

309 

.7759 

136 

37 

24 

562 

070 

.8495 

965 

36 

24 

090 

347 

.7747 

121 

36 

25 

.47588 

.54107 

1.8482 

.87951 

35 

25 

.49116 

.56385 

1.7735 

.87107 

35 

26 

614 

145 

.8469 

937 

34 

26 

141 

424 

.7723 

093 

34 

27 

639 

183 

.8456 

923 

33 

27 

166 

462 

.7711 

079 

33 

28 

6a5 

220 

.8443 

909 

32 

28 

192 

501 

.7699 

064 

32 

29 

690 

258 

.8430 

896 

31 

29 

217 

539 

.7687 

050 

31 

30 

.47716 

.54296 

1.8418 

.87882 

30 

30 

.49242 

.56577 

1.7675 

.87036 

30 

31 

741 

333 

.8405 

868 

29 

31 

268 

616 

.7663 

021 

29 

32 

767 

371 

.8392 

854 

28 

32 

293 

654 

.7651 

.87007 

28 

33 

793 

409 

.8379 

840 

27 

33 

318 

693 

.7639 

.86993 

27 

34 

818 

446 

.8367 

826 

26 

34 

344 

731 

.7627 

978 

26 

35 

.47844 

.54484 

1.8354 

.87812 

25 

35 

.49369 

.56769 

1.7615 

.86964 

25 

36 

869 

522 

.8341 

798 

24 

36 

394 

808 

.7603 

949 

24 

37 

895 

560 

.8329 

784 

23 

37 

419 

846 

.7591 

935 

23 

38 

920 

597 

.8316 

770 

22 

38 

445 

885 

.7579 

921 

22 

39 

946 

635 

.8303 

756 

21 

39 

470 

923 

.7567 

906 

21 

40 

.47971 

.54673 

1.8291 

.87743 

20 

40 

.49195 

.56962 

1.7556 

.86892 

20 

41 

.47997 

711 

.8278 

729 

19 

41 

521 

.57000 

.7544 

878 

19 

42 

.48022 

748 

.8265 

715 

18 

42 

546 

039 

.7532 

863 

18 

43 

048 

786 

.8253 

701 

17. 

43 

571 

078 

.7520 

849 

17 

44 

073 

824 

.8240 

687 

16 

44 

596 

116 

.7508 

834 

16 

45 

.48099 

.54862 

1.8228 

.87673 

15 

45 

.49622 

.57155 

1.7496 

.86820 

15 

46 

124 

900 

.8215 

659 

14 

46 

647 

193 

.7485 

805 

14 

47 

150 

938 

.8202 

645 

13 

47 

672 

232 

.7473 

791 

13 

48 

175 

.54975 

.8190 

631 

12 

48 

697 

271 

.7461 

777 

12 

49 

201 

.55013 

.8177 

617 

11 

49 

723 

309 

.7449 

762 

11 

50 

.48226 

.55051 

1.8165 

.87603 

10 

50 

.49748 

.57348 

1.7437 

.86748 

10 

51 

252 

089 

.8152 

589 

9 

51 

773 

386 

.7426 

733 

9 

52 

277 

127 

.8140 

575 

8 

52 

798 

425 

.7414 

719 

8 

53 

303 

165 

.8127 

561 

7 

53 

824 

464 

.7402 

704 

7 

54 

328 

203 

.8115 

546 

6 

54 

849 

503 

.7391 

690 

6 

55 

.48354 

.55241 

1.8103 

.87532 

5 

55 

.49874 

.57541 

1.7379 

.86675 

5 

56 

379 

279 

.8090 

518 

4 

56 

899 

580 

.7367 

661 

4 

57 

405 

317 

.8078 

504 

3 

57 

924 

619 

.7355 

646 

3 

58 

430 

355 

.8065 

490 

2 

58 

950 

657 

.7344 

632 

2 

59 

456 

393 

.8053 

476 

1 

59 

.49975 

696 

.7332 

617 

1 

60 

.48481 

.55431 

1.8040 

.87462 

0 

60 

.50000 

.57735 

1.7321 

.86603 

0 

Cos 

Ctn 

Tan 

Sin 

/ 

Cos 

Ctn 

Tan 

Sin 

/ 

fir 


fiO° 


II] 

30"— Values  of  Trigouometi 

•ic  Fuuctions  — 3r 

37 

1 

Sin 

Tan 

Ctn 

Cos 

! 

Sin 

Tan 

Ctn 

Cos 

0 

.50000 

.57735 

1.7321 

.86603 

60 

0 

.51504 

.60086 

1.6643 

.85717 

60 

1 

025 

774 

.7309 

588 

59 

1 

529 

126 

.6632 

702 

59 

2 

050 

813 

.7297 

573 

58 

2 

554 

165 

.6(321 

687 

58 

S 

076 

851 

.7286 

559 

57 

3 

579 

205 

.6610 

672 

57 

4 

101 

890 

.7274 

544 

56 

4 

604 

245 

.6599 

657 

56 

5 

.50126 

.57929 

1.7262 

.86530 

55 

5 

.51628 

.60284 

1.6588 

.85642 

55 

(> 

151 

.57i)68 

.7251 

515 

54 

6 

653 

324 

.6577 

627 

54 

7 

176 

.58007 

.7239 

501 

53 

7 

678 

364 

.65(36 

612 

53 

8 

201 

046 

.7228 

486 

52 

8 

703 

403 

.6555 

597 

52 

9 

227 

085 

.7216 

471 

51 

9 

728 

443 

.6545 

582 

51 

10 

.50252 

.58124 

1.7205 

.86457 

50 

10 

.51753 

.60483 

1.6534 

.85567 

50 

11 

277 

162 

.7193 

442 

49 

11 

778 

522 

.6523 

551 

49 

12 

302 

201 

.7182 

427 

48 

12 

803 

562 

.6512 

536 

48 

13 

327 

240 

.7170 

413 

47 

13 

828 

602 

.6501 

521 

47 

14 

352 

279 

.7159 

398 

4<3 

14 

852 

642 

.6490 

506 

46 

15 

.50377 

.58318 

1.7147 

.86384 

45 

15 

.51877 

.60681 

1.6479 

.85491 

45 

Ifi 

403 

357 

.71:36 

369 

44 

16 

902 

721 

.6469 

476 

44 

17 

428 

396 

.7124 

354 

43 

17 

927 

761 

.6458 

461 

43 

18 

453 

435 

.7113 

'MQ 

42 

18 

952 

801 

.6447 

446 

42 

19 

478 

474 

.7102 

325 

41 

19 

.51977 

841 

.6436 

431 

41 

20 

.50503 

.58513 

1.7090 

.86310 

40 

20 

.52002 

.60881 

1.6426 

.85416 

40 

21 

528 

552 

.7079 

295 

39 

21 

026 

921 

.6415 

401 

39 

22 

653 

591 

.7067 

281 

38 

22 

051 

.60960 

.(3404 

385 

(38 

23 

578 

631 

.7056 

266 

37 

23 

07(3 

.61000 

.6393 

370 

37 

24 

603 

670 

.7045 

251 

36 

24 

101 

040 

.6383 

355 

3(3 

25 

.50628 

.58709 

1.70:33 

.86237 

35 

25 

.52126 

.61080 

1.6372 

.85340 

35 

26 

654 

748 

.7022 

222 

34 

26 

151 

120 

.6361 

325 

54 

27 

679 

787 

.7011 

207 

33 

27 

175 

160 

.6351 

310 

3:3 

28 

704 

826 

.6999 

192 

32 

28 

200 

200 

.6340 

294 

32 

29 

729 

865 

.6988 

178 

31 

29 

225 

240 

.6329 

279 

31 

30 

.50754 

.58905 

1.6977 

.86163 

30 

30 

.52250 

.61280 

1.6319 

.85204 

30 

31 

779 

944 

.6965 

148 

29 

31 

275 

320 

.6308 

249 

29 

32 

804 

.58983 

.6954 

133 

28 

32 

299 

360 

.6297 

234 

28 

33 

829 

.59022 

.6^3 

119 

27 

33 

324 

400 

.6287 

218 

27 

34 

854 

061 

.6932 

104 

26 

34 

349 

440 

.6276 

203 

26 

35 

.50879 

.59101 

1.6920 

.86089 

25 

35 

.52374 

.61480 

1.6265 

.85188 

25 

3t3 

904 

140 

.6909 

074 

24 

36 

399 

520 

.6255 

173 

24 

37 

929 

179 

.6898 

059 

23 

37 

423 

561 

.6244 

157 

23 

38 

954 

218 

.6887 

045 

22 

38 

448 

601 

.6234 

142 

22 

39 

.50979 

258 

.6875 

0:30 

21 

39 

473 

641 

.6223 

127 

21 

40 

.51004 

.59297 

1.6864 

.8()015 

20 

40 

.52498 

.61681 

1.6212 

.85112 

20 

41 

029 

336 

.6853 

.8(3000 

19 

41 

522 

721 

.6202 

096 

19 

42 

054 

376 

.6J^2 

.85985 

18 

42 

547 

761 

.6191 

081 

18 

43 

079 

415 

.6831 

970 

17 

43 

572 

801 

.6181 

066 

17 

44 

104 

454 

.6820 

956 

16 

44 

597 

842 

.6170 

051 

16 

45 

.51129 

.59494 

1.6808 

.85941 

15 

45 

.52621 

.61882 

1.6160 

.85035 

15 

4(5 

154 

533 

.6797 

926 

14 

46 

646 

922 

.6149 

020 

14 

47 

179 

573 

.6786 

911 

13 

47 

671 

.61962 

.6139 

.85005 

13 

48 

204 

612 

.6775 

896 

12 

48 

696 

.62003 

.6128 

.84989 

12 

49 

229 

651 

.6764 

881 

11 

49 

720 

043 

.6118 

974 

11 

50 

.51254 

.59691 

1.6753 

.85866 

10 

50 

.52745 

.62083 

1.6107 

.84959 

10 

51 

279 

730 

.6742 

851 

9 

51 

770 

124 

.6097 

943 

9 

52 

304 

770 

.6731 

836 

8 

52 

794 

164 

.6087 

928 

8 

53 

329 

809 

.6720 

821 

7 

53 

819 

204 

.6076 

913 

7 

54 

354 

849 

.6709 

806 

6 

54 

844 

245 

.6066 

897 

6 

55 

.51379 

.59888 

1.6698 

.85792 

5 

55 

.52869 

.62285 

1.6055 

.84882 

5 

56 

404 

928 

.6687 

777 

4 

56 

893 

325 

.6045 

866 

4 

57 

429 

.59967 

.6676 

762 

3 

57 

918 

366 

.6034 

851 

3 

58 

454 

.60007 

.6665 

747 

2 

58 

943 

406 

.6024 

836 

2 

59 

479 

046 

.6654 

732 

1 

59 

967 

446 

.6014 

820 

1 

60 

.51504 

.60086 

1.6643 

.85717 

0 

60 

.52992 

.62487 

1.6003 

.84805 

0 

Cos 

Ctn  !  Tan 

Sin 

/ 

Cos 

Ctn 

Tan 

Sin 

1 

59° 


68° 


38 

32°  — Values  of  Trigonometric  Functions  —  33° 

[ir 

/ 

Sin 

Tan 

Ctn 

Cos 

/ 

Sin 

Tan 

Ctn 

Cos 

0 

.52992 

.62487 

1.6003 

.84805 

60 

0 

.54464 

.64941 

1.5399 

.83867 

60 

1 

.53017 

527 

.5993 

789 

59 

1 

488 

.64982 

.5389 

851 

59 

2 

041 

568 

.5983 

774 

58 

2 

513 

.65024 

.5379 

835 

58 

3 

066 

608 

.5972 

759 

57 

3 

537 

065 

.5369 

819 

57 

4 

091 

649 

.5962 

743 

56 

4 

561 

106 

.5359 

804 

56 

5 

.53115 

.62689 

1.5952 

.84728 

55 

5 

.54586 

.65148 

1.5350 

.83788 

55 

6 

140 

730 

.5941 

712 

54 

6 

610 

189 

.5340 

772 

54 

7 

164 

770 

.5931 

697 

53 

7 

635 

231 

.5330 

756 

53 

8 

189 

811 

.5921 

681 

52 

8 

659 

272 

.5320 

740 

52 

9 

214 

852 

.5911 

666 

51 

9 

683 

314 

.5311 

724 

51 

10 

.53238 

.62892 

1.5900 

.84650 

50 

10 

.54708 

.65355 

1.5301 

.83708 

50 

11 

263 

933 

.5890 

635 

49 

11 

732 

397 

.5291 

692 

49 

12 

288 

.62973 

.5880 

619 

48 

12 

756 

438 

.5282 

676 

48 

13 

312 

.63014 

.5869 

604 

47 

13 

781 

480 

.5272 

660 

47 

14 

337 

055 

.5859 

588 

46 

14 

805 

521 

.5262 

645 

46 

15 

.53361 

.63095 

1.5849 

.84573 

45 

15 

.54829 

.65563 

1.5253 

.83629 

45 

16 

386 

136 

.5839 

557 

44 

16 

854 

604 

.5243 

613 

44 

17 

411 

177 

.5829 

542 

43 

17 

878 

646 

.5233 

597 

43 

18 

435 

217 

.5818 

526 

42 

18 

902 

688 

.5224 

581 

42 

19 

460 

258 

.5808 

511 

41 

19 

927 

729 

.5214 

565 

41 

20 

.53484 

.63299 

1.5798 

.84495 

40 

20 

.54951 

.65771 

1.5204 

.83549 

40 

21 

509 

340 

.5788 

480 

39 

21 

975 

813 

.5195 

533 

39 

22 

534 

380 

.5778 

464 

38 

22 

.54999 

854 

.5185 

617 

38 

23 

558 

421 

.5768 

448 

37 

23 

.55024 

896 

.5175 

501 

37 

24 

683 

462 

.5757 

433 

36 

24 

048 

938 

.5166 

485 

36 

25 

.53607 

.63503 

1.5747 

.84417 

35 

25 

.55072 

.65980 

1.5156 

.83469 

35 

26 

632 

544 

.5737 

402 

34 

26 

097 

.66021 

.5147 

453 

34 

27 

656 

584 

.5727 

386 

33 

27 

121 

063 

.5137 

437 

33 

28 

681 

625 

.5717 

370 

32 

28 

145 

105 

.5127 

421 

32 

29 

705 

666 

.5707 

355 

31 

29 

169 

147 

.5118 

405 

31 

30 

.53730 

.63707 

1.5697 

.84339 

30 

30 

.55194 

.66189 

1.5108 

.83389 

30 

31 

754 

748 

.5687 

324 

29 

31 

218 

230 

.5099 

373 

29 

32 

779 

789 

.5677 

308 

28 

32 

242 

272 

.5089 

356 

28 

33 

804 

830 

.5667 

292 

27 

33 

266 

314 

.5080 

340 

27 

34 

828 

871 

.5657 

277 

26 

34 

291 

356 

.5070 

324 

26 

35 

.53853 

.63912 

1.5647 

.84261 

25 

35 

.55315 

.66398 

1.5061 

.83308 

25 

36 

877 

953 

.5637 

245 

24 

36 

339 

440 

.5051 

292 

24 

37 

902 

.63994 

.5627 

230 

23 

37 

363 

482 

.5042 

276 

23 

38 

926 

.64035 

.5617 

214 

22 

38 

388 

524 

.5032 

260 

22' 

39 

951 

076 

.5607 

198 

21 

39 

412 

566 

.5023 

244 

21 

40 

.53975 

.64117 

1.5597 

.84182 

20 

40 

.55436 

.66608 

1.5013 

.83228 

20 

41 

.54000 

158 

.5587 

167 

19 

41 

460 

650 

.5004 

212 

19 

42 

024 

199 

.5577 

151 

18 

42 

484 

692 

.4994 

195 

18 

43 

049 

240 

.5567 

135 

17 

43 

509 

734 

.4985 

179 

17 

44 

073 

281 

.5557 

120 

16 

44 

533 

776 

.4975 

163 

16 

45 

.54097 

.64322 

1.5547 

.84104 

15 

45 

.55557 

.66818 

1.49(36 

.83147 

15 

46 

122 

363 

.5537 

088 

14 

46 

581 

860 

.4957 

131 

14 

47 

146 

404 

.5527 

072 

13 

47 

605 

902 

.4947 

115 

13 

48 

171 

446 

.5517 

057 

12 

48 

630 

944 

.4938 

098 

12 

49 

195 

487 

.5507 

041 

11 

49 

654 

.66986 

.4928 

082 

11 

50 

.54220 

.64528 

1.5497 

.84025 

10 

50 

.55678 

.67028 

1.4919 

.83066 

10 

51 

244 

569 

.5487 

.84009 

9 

51 

702 

071 

.4910 

050 

9 

52 

269 

610 

.5477 

.83994 

8 

52 

726 

113 

.4900 

034 

8 

53 

293 

652 

.5468 

978 

7 

53 

750 

155 

.4891 

017 

7 

54 

317 

693 

.5458 

962 

6 

54 

775 

197 

.4882 

.83001 

6 

55 

.54342 

.64734 

1.5448 

.83946 

5 

55 

.55799 

.67239 

1.4872 

.82985 

5 

56 

366 

775 

.5438 

930 

4 

56 

823 

282 

.4863 

969 

4 

57 

391 

817 

.5428 

915 

3 

57 

847 

324 

.4854 

953 

3 

58 

415 

858 

.5418 

899 

2 

58 

871 

366 

.4844 

936 

2 

59 

440 

899 

.5408 

883 

1 

59 

895 

409 

.4835 

920 

1 

60 

.54464 

.64941 

1.5399 

.83867 

0 

60 

.55919 

.67451 

1.4826 

.82904 

0 

Cos 

Ctn 

Tan 

Sin 

/ 

Cos 

Ctn 

Tan 

Sin 

/ 

II] 


34°— Values  of  Trigonometric  Functions  — 35° 


39 


/ 

Sin 

Tan 

Gtn 

Cos 

0 

.55919 

.67451 

1.4826 

.82904 

60 

1 

943 

493 

.4816 

887 

59 

2 

968 

536 

.4807 

871 

58 

3 

.55992 

5'78 

.4798 

855 

57 

4 

.56016 

620 

.4788 

839 

56 

5 

.56040 

.67663 

1.4779 

.82822 

55 

() 

064 

705 

.4770 

806 

54 

7 

088 

748 

.4761 

790 

53 

8 

112 

790 

.4751 

773 

52 

9 

136 

832 

.4742 

757 

51 

10 

.56160 

.67875 

1.4733 

.82741 

50 

11 

184 

917 

.4724 

724 

49 

12 

208 

.67960 

.4715 

708 

48 

13 

232 

.68002 

.4705 

692 

47 

14 

256 

045 

.4696 

675 

46 

15 

.56280 

.68088 

1.4687 

.82659 

45 

16 

305 

130 

.4678 

643 

44 

17 

329 

173 

.4669 

626 

43 

18 

353 

215 

.4659 

610 

42 

19 

377 

258 

.4650 

593 

41 

20 

.56401 

.68301 

1.4641 

.82577 

40 

21 

425 

;^3 

.4632 

561 

39 

22 

449 

386 

.4623 

544 

38 

23 

473 

429 

.4614 

528 

37 

24 

497 

471 

.4605 

511 

36 

25 

.56521 

.68514 

1.4596 

.82495 

35 

26 

545 

557 

.4586 

478 

34 

27 

569 

600 

.4577 

462 

33 

28 

593 

642 

•4568 

446 

32 

29 

617 

685 

.4559 

429 

31 

30 

.56641 

.68728 

1.4550 

.82413 

30 

31 

.  665 

771 

.4541 

396 

29 

32 

689 

814 

.4532 

380 

28 

33 

713 

857 

.4523 

363 

27 

34 

736 

900 

.4514 

347 

26 

35 

.56760 

.68942 

1.4505 

.82330 

25 

36 

784 

.68985 

.4496 

314 

24 

37 

808 

.69028 

.4487 

297 

23 

38 

832 

071 

.4478 

281 

22 

39 

856 

114 

.4469 

264 

21 

40 

.56880 

.69157 

1.4460 

.82248 

20 

41 

904 

200 

.4451 

231 

19 

42 

928 

243 

.4442 

214 

18 

43 

952 

286 

.4433 

198 

17 

44 

.56976 

329 

.4424 

181 

16 

45 

.57000 

.69372 

1.4415 

.82165 

15 

46 

024 

416 

.4406 

148 

14 

47 

047 

459 

.4397 

132 

13 

48 

071 

502 

.4388 

115 

12 

49 

095 

545 

.4379 

098 

11 

50 

.57119 

.69588 

1.4370 

.82082 

10 

51 

143 

631 

.4361 

065 

9 

52 

167 

675 

.4352 

048 

8 

53 

191 

718 

.4344 

032 

7 

54 

215 

761 

.4335 

.82015 

6 

55 

.57238 

.69804 

1.4326 

.81999 

5 

.'56 

262 

847 

.4317 

982 

4 

57 

286 

891 

.4308 

965 

3 

58 

310 

934 

.4299 

949 

2 

59 

334 

.69977 

.4290 

932 

1 

60 

.57358 

.70021 

1.4281 

.81915 

0 

Cos 

Ctn 

Tan 

Sin 

/ 

Sin       Tan       Ctn 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 

60  .58779 
Cos 


.57358 
381 
405 
429 
453 

.57477 
501 
524 
548 
572 

.57596 
619 
643 
667 
691 

.57715 
738 
762 
786 
810 

.57833 
857 
881 
904 
928 

.57952 
976 

.57999 

.58023 
047 

.58070 
094 
118 
141 
165 

.58189 
212 
236 
260 
283 

.58307 
330 
354 
378 
401 

.58425 
449 
472 
496 
519 

.58543 
567 
590 
614 
637 

.58661 
684 
708 
731 
755 


.70021 
0()4 
107 
151 
194 

.70238 
281 
325 
368 
412 

.70455 
499 
542 
586 
629 

.70673 
717 
760 
804 
848 

.70891 
935 

.70979 

.71023 
066 

.71110 
154 
198 
242 
285 

.71329 
373 
417 
461 
505 

.71549 
593 
637 
681 
725 

.71769 
813 
857 
901 
946 

.71990 

.72034 
078 
122 
167 

.72211 
255 
299 
344 
388 

.72432 
477 
521 
565 
610 

.72654 


1.4281 
.4273 
.4264 
.4255 
.4246 

1.4237 
.4229 
.4220 
.4211 
.4202 

1.4193 
.4185 
.4176 
.4167 
.4158 

1.4150 
.4141 
.4132 
.4124 
.4115 

1.4106 
.4097 
.4089 
.4080 
.4071 

1.4063 
.4054 
.4045 
.4037 
.4028 

1.4019 
.4011 
.4002 
.3994 
.3985 

1.3976 
.39(38 
.3959 
.3951 
.3942 

1.3934 
.3925 
.3916 
.3908 
.3899 

1.3891 
.3882 
.3874 
.3865 
.3857 

1.3848 
.3840 
.3831 
.3823 
.3814 

1.3806 
.3798 
.3789 
.3781 
.3772 

1.3764 


Cos 


Ctn   Tan 


.81915 
899 
882 
865 
848 

.81832 
815 
798 
782 
765 

.81748 
731 
714 
698 
681 

.81664 
647 
631 
614 
597 

.81580 
563 
546 
530 
513 

.81496 
479 
462 
445 
428 

.81412 
395 
378 
361 
344 

.81327 
310 
293 
276 
259 

.81242 
225 
208 
191 
174 

.81157 
140 
123 
106 
089 

.81072 
055 
038 
021 

.81004 

.80987 
970 
953 
936 
919 

.80902 
Sin 


40 


36°  — Values  of  Trigonometric  Functions  —  37° 


0 

1 

2 
3 
4 

5 

6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 

40 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


Sin   Tan   Gtn 


.58779 
802 
826 
849 
873 

.58896 
920 
943 
967 

.58990 

.59014 
037 
061 
084 
108 

.59131 
154 
178 
201 
225 

.59248 
272 
295 
318 
342 

.59365 
389 
412 
436 
459 

.59482 
506 
529 
552 
576 

.59599 
622 
646 
669 
693 

.59716 
739 
763 
786 
809 

.59832 
856 
879 
902 
926 

.59949 
972 

.59995 

.60019 
042 

.60065 
089 
112 
135 
158 

.60182 


.72654 
699 
743 

788 
832 

.72877 
921 

.72966 

.73010 
055 

.73100 
144 
189 
234 
278 

.73323 
368 
413 
457 
502 

.73547 
592 
637 
681 
726 

.73771 
816 
861 
906 
951 

.73996 

.74041 
086 
131 
176 

.74221 
267 
312 
357 
402 

.74447 
492 
538 
583 
628 

.74674 
719 
764 
810 
855 

.74900 
946 

.74991 

.75037 
082 

.75128 
173 
219 
264 
310 

.75355 


Cos        Ctn       Tan 


1.3764 

.3755 
.3747 
.3739 
.3730 

1.3722 
.3713 
.3705 
.3697 
.3688 

1.3680 
.3672 
.:3663 
.3655 
.3647 

1.3638 
.3630 
.3(322 
.3613 
.3605 

1.3.597 
.3588 
.3580 
.3572 
.3564 

1.3555 
.3547 
.3539 
.3531 
.3522 

1.3514 
.3506 
.3498 
.3490 
.3481 

1.3473 
.3465 
.3457 
.3449 
.3440 

1.3432 
.3424 
.3416 
.3408 
.3400 

1.3392 
.3384 
.3375 
.3367 
.3359 

1.3351 
.3343 
.3335 
.3327 
.3319 

1.3311 
.3303 
.3295 
.3287 
.3278 

1.3270 


Cos 


.80902 
885 
867 
850 
833 

.80816 
799 
782 
765 
748 


.80730 
713 


50 

49 
48 
679  47 
662  46 


.80644 
627 
610 
593 
576 

.80558 
541 
524 
507 
489 

.80472 
455 
438 
420 
403 

.80386 
368 
351 
334 
316 

.80299 
282 
264 
247 
230 

.80212 
195 
178 
160 
143 

.80125 
108 
091 
073 
056 

.80038 
021 

.80003 

.79986 
968 

.79951 
934 
916 
899 
881 

.79864 


Sin 


/ 

Sin 

Tan 

Ctn 

Cos 

0 

.60182 

.75355 

1.3270 

.79864 

60 

1 

205 

401 

.3262 

846 

59 

2 

228 

447 

.3254 

829 

58 

3 

251 

492 

.3246 

811 

57 

4 

274 

538 

.3238 

793 

56 

5 

.60298 

.75584 

1.3230 

.79776 

55 

6 

321 

629 

.3222 

758 

54 

7 

344 

675 

.3214 

741 

53 

8 

367 

721 

.3206 

723 

52 

9 

390 

767 

.3198 

706 

51 

10 

.60414 

.75812 

1.3190 

.79688 

50 

11 

437 

858 

.3182 

671 

49 

12 

460 

904 

.3175 

653 

48 

13 

483 

950 

.3167 

635 

47 

14 

506 

.75996 

.3159 

618 

46 

15 

.60529 

.76042 

1.3151 

.79600 

45 

16 

553 

088 

.3143 

583 

44 

17 

576 

134 

.3135 

565 

43 

18 

599 

180 

.3127 

547 

42 

19 

622 

226 

.3119 

630 

41 

20 

.60645 

.76272 

1.3111 

.79512 

40 

21 

668 

318 

.3103 

494 

39 

22 

691 

364 

.3095 

477 

38 

23 

714 

410 

.3087 

459 

37 

24 

738 

456 

.3079 

441 

36 

25 

.60761 

.76502 

1.3072 

.79424 

85 

26 

784 

548 

.3064 

406 

34 

27 

807 

594 

.3056 

388 

33 

28 

830 

640 

.3048 

371 

32 

29 

853 

686 

.3040 

353 

31 

30 

.60876 

.76733 

1.3032 

.79335 

30 

31 

899 

779 

.3024 

318 

29 

32 

922 

825 

.3017 

300 

28 

33 

945 

871 

.3009 

282 

27 

34 

968 

918 

.3001 

264 

26 

35 

.60991 

.76f)64 

1.2993 

.79247 

25 

36 

.61015 

.77010 

.2985 

229 

24 

37 

038 

057 

.2977 

211 

23 

38 

061 

103 

.2970 

193 

22 

39 

084 

149 

.2962 

17() 

21 

40 

.61107 

.77196 

1.2954 

.79158 

20 

41 

130 

242 

.2946 

140 

19 

42 

153 

289 

.2938 

122 

18 

43 

176 

335 

.2931 

105 

17 

44 

199 

382 

.2923 

087 

16 

45 

.61222 

.77428 

1.2915 

.79069 

15 

46 

245 

475 

.2907 

051 

14 

47 

268 

521 

.2900 

033 

13 

48 

291 

568 

.2892 

.79016 

12 

49 

314 

615 

.2884 

.78993 

11 

60 

.61.337 

.77661 

1.2876 

.78980 

10 

51 

360 

708 

.2869 

962 

9 

52 

383 

754 

.2861 

944 

8 

53 

406 

801 

.2853 

926 

7 

54 

429 

848 

.2846 

908 

6 

55 

.61451 

.77895 

1.2838 

.78891 

5 

56 

474 

941 

.2830 

873 

4 

57 

497 

.77988 

.2822 

855 

3 

58 

520 

.78035 

.2815 

837 

2 

59 

543 

082 

.2807 

819 

1 

60 

.61566 

.78129 

1.2799 

.78801 

0 

Cos 

Ctn 

Tan 

Sin 

1 

n] 

38°— Values  of  Trigonometric  Functions  —  39° 

41 

/ 

Sin 

Tan 

Ctn 

Cos 

f 

Sin 

Tan 

Ctn 

Cos 

0 

.61566 

.78129 

1.279f) 

.78801 

60 

0 

.62932 

.80978 

1.2349 

.77715 

60 

1 

589 

175 

.2792 

783 

59 

1 

955 

.81027 

.2342 

696 

59 

2 

612 

222 

.2784 

765 

58 

o 

.62977 

075 

.2334 

678 

58 

3 

635 

269 

.2776 

747 

57 

3 

.63000 

123 

.2327 

660 

57 

4 

658 

316 

.2769 

729 

56 

4 

022 

171 

.2320 

641 

56 

5 

.61681 

.78363 

1.2761 

.78711 

55 

5 

.63045 

.81220 

1.2312 

.77623 

55 

6 

704 

410 

.2753 

694 

54 

6 

068 

268 

.2305 

605 

54 

7 

726 

457 

.2746 

676 

53 

7 

090 

316 

.2298 

586 

53 

S 

749 

504 

.2738 

658 

52 

8 

113 

364 

.2290 

568 

52 

9 

772 

551 

.2731 

640 

51 

9 

135 

413 

.2283 

550 

51 

10 

.61795 

.78598 

1.2723 

.78622 

50 

10 

.63158 

.81461 

1.2276 

.77531 

50 

11 

818 

645 

.2715 

604 

49 

11 

180 

510 

.2268 

513 

49 

12 

841 

692 

.2708 

586 

48 

12 

203 

558 

.2261 

494 

48 

13 

864 

739 

.2700 

568 

47 

13 

225 

606 

.2254 

476 

47 

14 

887 

786 

.2693 

550 

46 

14 

248 

655 

.2247 

458 

46 

15 

.61909 

.78834 

1.2685 

.78532 

45 

15 

.63271 

.81703 

1.2239 

.77439 

45 

16 

932 

881 

.2677 

514 

44 

16 

293 

752 

.2232 

421 

44 

17 

955 

928 

.2670 

496 

43 

17 

316 

800 

.2225 

402 

43 

18 

.61978 

.78975 

.2662 

478 

42 

18 

338 

849 

.2218 

384 

42 

19 

.62001 

.79022 

.2655 

460 

41 

19 

361 

898 

.2210 

366 

41 

20 

.62024 

.79070 

1.2647 

.78442 

40 

20 

.63383 

.81946 

1.2203 

.77347 

40 

21 

046 

117 

.2640 

424 

39 

21 

406 

.81995 

.2196 

329 

39 

22 

069 

164 

.2632 

405 

38 

22 

428 

.82044 

.2189 

310 

38 

23 

092 

212 

.2624 

387 

37 

23 

451 

092 

.2181 

292 

37 

24 

115 

259 

.2617 

369 

36 

24 

473 

141 

.2174 

273 

36 

25 

.62138 

.79306 

1.2609 

.78351 

35 

25 

.63496 

.82190 

1.2167 

.77255 

35 

26 

160 

354 

.2602 

333 

34 

26 

518 

238 

.2160 

236 

34 

27 

183 

401 

.2594 

315 

33 

27 

540 

287 

.2153 

218 

33 

28 

206 

449 

.2587 

297 

32 

28 

563 

336 

.2145 

199 

32 

29 

229 

496 

.2579 

279 

31 

29 

585 

385 

.2138 

181 

31 

30 

.62251 

.79544 

1.2572 

.78261 

30 

30 

.63608 

.82434 

1.2131 

.77162 

30 

31 

274 

591 

.2564 

243 

29 

31 

630 

483 

.2124 

144 

29 

32 

297 

639 

.2557 

225 

28 

32 

653 

531 

.2117 

125 

28 

33 

320 

686 

.2549 

206 

27 

33 

675 

680 

.2109 

107 

27 

34 

342 

734 

.2542 

188 

26 

34 

698 

629 

.2102 

088 

26 

35 

.62365 

.79781 

1.25:34 

.78170 

25 

35 

.63720 

.82678 

1.2095 

.77070 

25 

36 

388 

829 

.2527 

152 

24 

36 

742 

727 

.2088 

051 

24 

37 

411 

877 

.2519 

134 

23 

37 

765 

776 

.2081 

033 

23 

38 

433 

924 

.2512 

116 

22 

38 

787 

825 

.2074 

.77014 

22 

39 

456 

.79972 

.2504 

098 

21 

39 

810 

874 

.2066 

.76996 

21 

40 

.62479 

.80020 

1.2497 

.78079 

20 

40 

.63832 

.82923 

1.2059 

.76977 

20 

41 

502 

067 

.2489 

061 

19 

41 

854 

.82972 

.2052 

959 

19 

42 

524 

115 

.2482 

043 

18 

42 

877 

.83022 

.2045 

940 

18 

43 

547 

163 

.2475 

025 

17 

43 

899 

071 

.2038 

921 

17 

44 

570 

211 

.2467 

.78007 

16 

44 

922 

120 

.2031 

903 

16 

45 

.62592 

.80258 

1.2460 

.77988 

15 

45 

.63944 

.83169 

1.2024 

.76884 

15 

46 

615 

306 

.2452 

970 

14 

46 

966 

218 

.2017 

866 

14 

47 

638 

354 

.2445 

952 

13 

47 

.63989 

268 

.2009 

847 

13 

48 

660 

402 

.2437 

934 

12 

48 

.64011 

317 

.2002 

828 

12 

49 

683 

450 

.2430 

916 

11 

49 

033 

366 

.1995 

810 

11 

50 

.62706 

.80498 

1.2423 

.77897 

10 

50 

.64056 

.83415 

1.1988 

.76791 

10 

51 

728 

546 

.2415 

879 

9 

51 

078 

465 

.1981 

772 

9 

52 

751 

594 

.2408 

861 

8 

52 

100 

514 

.1974 

754 

8 

53 

774 

642 

.2401 

843 

7 

53 

123 

564 

.1967 

735 

7 

54 

796 

690 

.2393 

824 

6 

54 

145 

613 

.1960 

717 

6 

55 

.62819 

.80738 

1.2386 

.77806 

5 

55 

.64167 

.83662 

1.1953 

.76698 

5 

56 

842 

786 

.2378 

788 

4 

56 

190 

712 

.1946 

679 

4 

57 

864 

834 

.2371 

769 

3 

57 

212 

761 

.1939 

661 

3 

58 

887 

882 

.2364 

751 

2 

58 

234 

811 

.1932 

642 

2 

59 

909 

930 

.2356 

733 

1 

59 

256 

860 

.1925 

623 

1 

60 

.62932 

.80978 

1.2349 

.77715 

0 

60 

.64279 

.83910 

1.1918 

.76604 

0 

Cos 

Ctn 

Tan 

Sin 

' 

Cos 

Ctn 

Tan 

Sin 

/ 

^V 


^II0 


42 

40°  — Values  of  Trigonometric  Functions  — 41° 

[n 

/ 

Sin 

Tan 

Ctn 

Cos 

/ 

Sin 

Tan 

Ctn 

Cos 

0 

.64279 

.83910 

1.1918 

.76604 

60 

0 

.65606 

.86929 

1.1504 

.75471 

60 

1 

301 

.83960 

.1910 

586 

59 

1 

628 

.86980 

.1497 

452 

59 

2 

323 

.84009 

.1903 

667 

58 

2 

650 

.87031 

.1490 

433 

58 

3 

346 

059 

.1896 

648 

57 

3 

672 

082 

,1483 

414 

57 

4 

368 

108 

.1889 

530 

56 

4 

694 

1.33 

.1477 

395 

56 

5 

.64390 

.84158 

1.1882 

.76511 

55 

5 

.65716 

.87184 

1.1470 

.75375 

55 

6 

412 

208 

.1875 

492 

54 

6 

738 

236 

.14(J3 

356 

54 

7 

435 

258 

.1868 

473 

53 

7 

759 

287 

.1456 

337 

53 

8 

457 

307 

.1861 

455 

52 

8 

781 

338 

.1450 

318 

52 

9 

479 

357 

.1854 

436 

51 

9 

803 

389 

.1443 

299 

51 

10 

.64501 

.84407 

1.1847 

.76417 

50 

10 

.65825 

.87441 

1.1436 

.75280 

60 

11 

624 

457 

.1840 

398 

49 

11 

847 

492 

.1430 

261 

49 

12 

646 

507 

.1833 

380 

48 

12 

869 

543 

.1423 

241 

48 

13 

568 

556 

.1826 

361 

47 

13 

891 

595 

.1416 

222 

47 

14 

590 

606 

.1819 

342 

46 

14 

913 

646 

.1410 

203 

46 

15 

.64612 

.84656 

1.1812 

.T6323 

45 

15 

.65935 

.87698 

1.1403 

.75184 

45 

16 

635 

706 

.1806 

304 

44 

16 

956 

749 

.1396 

.  165 

44 

17 

657 

756 

.1799 

286 

43 

17 

.65978 

801 

.1389 

146 

43 

18 

679 

806 

.1792 

267 

42 

18 

.66000 

852 

.1383 

126 

42 

19 

701 

856 

.1785 

248 

41 

19 

022 

904 

.1376 

107 

41 

20 

.64723 

.84906 

1.1778 

.76229 

40 

20 

.66044 

.87955 

1.1369 

.75088 

40 

21 

746 

.84956 

.1771 

210 

39 

21 

066 

.88007 

.1363 

069 

39 

22 

768 

.85006 

.1764 

192 

38 

22 

088 

059 

.1356 

050 

38 

23 

790 

057 

.1757 

173 

37 

23 

109 

110- 

.1349 

030 

37 

24 

812 

107 

.1750 

154 

36 

24 

131 

162 

.1343 

.75011 

36 

25 

.64834 

.85157 

1.1743 

.76135 

35 

25 

.66153 

.88214 

1.1336 

.74992 

35 

26 

856 

207 

.1736 

116 

34 

26 

175 

265 

.1329 

973 

34 

27 

878 

257 

.1729 

097 

33 

27 

197 

317 

.1323 

953 

33 

28 

901 

308 

.1722 

078 

32 

28 

218 

369 

.1316 

934 

32 

29 

923 

358 

.1715 

059 

31 

29 

240 

421 

.1310 

915 

31 

30 

.64945 

.85408 

1.1708 

.76041 

30 

30 

.66262 

.88473 

1.1303 

.74896 

30 

31 

967 

458 

.1702 

022 

29 

31 

284 

524 

.1296 

876 

29 

32 

.()4989 

509 

.1695 

.76003 

28 

32 

306 

576 

.1290 

857 

28 

33 

.65011 

559 

.1688 

.75984 

27 

33 

327 

628 

.1283 

838 

27 

34 

033 

609 

.1681 

9()5 

26 

34 

349 

680 

.1276 

818 

26 

35 

.65055 

.85660 

1.1674 

.75946 

25 

35 

.66371 

.88732 

1.1270 

.74799 

25 

36 

077 

710 

.1667 

927 

24 

36 

393 

784 

.1263 

780 

24 

37 

100 

761 

.1660 

908 

23 

37 

414 

836 

.1257 

7(50 

23 

38 

122 

811 

.1653 

889 

22 

38 

436 

888 

.1250 

741 

22 

39 

144 

862 

.1647 

870 

21 

39 

458 

940 

.1243 

722 

21 

40 

.65166 

.85912 

1.1640 

.75851 

20 

40 

.66480 

.88992 

1.1237 

.74703 

20 

41 

188 

.85963 

.1633 

832 

19 

41 

501 

.89045 

.1230 

683 

19 

42 

210 

.86014 

.1626 

813 

18 

42 

623 

097 

.1224 

664 

18 

43 

232 

064 

.1619 

794 

17 

43 

545 

149 

.1217 

644 

17 

44 

254 

115 

.1612 

775 

16 

44 

566 

201 

.1211 

625 

16 

45 

.65276 

.86166 

1.1606 

.75756 

15 

45 

.66588 

.89253 

1.1204 

.74606 

15 

46 

298 

216 

.1599 

738 

14 

46 

610 

306 

.1197 

586 

14 

47 

320 

267 

.1592 

719 

13 

47 

632 

358 

.1191 

567 

13 

48 

342 

318 

.1585 

700 

12 

48 

653 

410 

.1184 

548 

12 

49 

364 

368 

.1578 

680 

11 

49 

675 

463 

.1178 

628 

11 

50 

.65386 

.86419 

1.1571 

.75661 

10 

50 

.66697 

.89516 

1.1171 

.74509 

10 

51 

408 

470 

.1565 

642 

9 

51 

718 

567 

.1166 

489 

9 

52 

430 

521 

.1558 

623 

8 

52 

740 

620 

.1158 

470 

8 

53 

452 

572 

.1551 

604 

7 

53 

762 

672 

.1152 

451 

7 

54 

474 

623 

.1544 

585 

6 

54 

783 

725 

.1145 

431 

6 

55 

.65496 

.86674 

1.1538 

.75566 

5 

55 

.66805 

.89777 

1.1139 

.74412 

5 

56 

518 

725 

.1531 

547 

4 

56 

827 

830 

.1132 

392 

4 

57 

540 

776 

.1524 

528 

3 

57 

848 

883 

.1126 

373 

3 

58 

562 

827 

.1517 

509 

2 

58 

870 

935 

.1119 

353 

2 

59 

584 

878 

.1510 

490 

1 

59 

891 

.89988 

.1113 

334 

1 

60 

.65606 

.86929 

1.1504 

.75471 

0 

60 

.66913 

•90010 

1.1106 

.74314 

0 

Cos 

Ctn 

Tan 

Sin 

/ 

Cos 

Ctn 

Tan 

Sin 

t 

49° 


48° 


42° — Values  of  Trigonometric  Functions  —  43° 


43 


0 

Sin 

1  Tan 

Ctn 

Cos 

f 

Sin 

Tan 

Ctn 

Cos 

.(5t)9i;J 

.1KM)40 

1.1106 

.74314 

60 

0 

.68200 

.93252 

1.0724 

.73135 

60 

1 

9;35 

093 

.1100 

295 

59 

1 

221 

306 

.0717 

116 

59 

2 

956 

146 

,1093 

276 

58 

2 

242 

360 

.0711 

096 

58 

3 

978 

199 

.1087 

256 

57 

3 

264 

415 

.0705 

076 

57 

4 

.66999 

251 

.1080 

237 

56 

4 

285 

469 

.0699 

056 

56 

5 

.67021 

.90304 

1.1074 

.74217 

55 

5 

.68306 

.93524 

1.0(592 

.73036 

55 

() 

043 

357 

.1067 

198 

54 

() 

327 

578 

.0686 

.73016 

54 

7 

064 

410 

.1061 

178 

53 

7 

349 

633 

.0680 

.72996 

53 

8 

08() 

463 

.1054 

159 

52 

8 

370 

688 

.0674 

976 

52 

9 

107 

516 

.1048 

139 

51 

9 

391 

742 

.0668 

957 

51 

10 

.67129 

.90569 

1.1041 

.74120 

50 

10 

.68412 

.93797 

1.0661 

.72937 

50 

11 

151 

621 

.ioa5 

100 

49 

11 

434 

852 

.0655 

917 

49 

12 

172 

674 

.1028 

080 

48 

12 

455 

906 

.0649 

897 

48 

13 

194 

727 

.1022 

061 

47 

13 

476 

.93961 

.0(543 

877 

47 

14 

215 

781 

.1016 

041 

46 

.  14 

497 

.94016 

.0637 

857 

46 

15 

.672.37 

.90834 

1.1009 

.74022 

45 

15 

.68518 

.94071 

1.0630 

.72837 

45 

1(> 

258 

887 

.1003 

.74002 

44 

16 

539 

125 

.0624 

817 

44 

17 

280 

940 

.0996 

.73983 

43 

17 

561 

180 

.0(518 

797 

43 

18 

301 

.90993 

.0990 

963 

42 

18 

582 

235 

.0612 

777 

42 

19 

323 

.91046 

.0983 

944 

41 

19 

603 

290 

.0606 

757 

41 

20 

.67344 

.91099 

1.0977 

.73924 

40 

20 

.68624 

.94345 

1.0599 

.72737 

40 

21 

366 

153 

.0971 

904 

39 

21 

645 

400 

.0593 

717 

39 

22 

387 

206 

.0964 

885 

38 

22 

666 

455 

.0587 

697 

38 

23 

409 

259 

.0958 

865 

37 

23 

688 

510 

.0581 

677 

37 

24 

430 

313 

.0951 

846 

36 

24 

709 

565 

.0575 

657 

36 

25 

.674.52 

.91366 

1.0945 

.73826 

35 

25 

.68730 

.^620 

1.0569 

.72637 

35 

2(J 

473 

419 

.0939 

806 

34 

2(5 

751 

676 

.05(52 

617 

34 

27 

495 

473 

.0932 

787 

33 

27 

772 

731 

.0556 

597 

33 

28 

516 

526 

.0926 

767 

32 

28 

793 

786 

.0550 

677 

32 

29 

538 

580 

.0919 

747 

31 

29 

814 

841 

.0544 

657 

31 

30 

.67559 

.91633 

1.0913 

.73728 

30 

30 

.68835 

.94896 

1.0538 

.72537 

30 

31 

580 

687 

.0907 

708 

29 

31 

857 

.94952 

.0532 

517 

29 

32 

(502 

740 

.0900 

(588 

28 

32 

878 

.95007 

.0526 

497 

28 

33 

623 

794 

.0894 

669 

27 

33 

899 

062 

.0519 

477 

27 

34 

(545 

847 

.0888 

649 

26 

34 

920 

118 

.0513 

457 

26 

35 

.67666 

.91901 

1.0881 

.73629 

25 

35 

.68941 

.95173 

1.0507 

.72437 

25 

3(3 

(588 

.91955 

.0875 

610 

24 

36 

%2 

229 

.0501 

417 

24 

37 

709 

.92008 

.0869 

590 

23 

37 

.68983 

284 

.0495 

397 

23 

38 

730 

062 

.0862 

570 

22 

38 

.69004 

340 

.0489 

377 

22 

39 

752 

116 

.0856 

551 

21 

39 

025 

395 

.0483 

357 

21 

40 

.67773 

.92170 

1.0850 

.73531 

20 

40 

.69046 

.95451 

1.0477 

.72337 

20 

41 

795 

224 

.0843 

511 

19 

41 

067 

506 

.0470 

317 

19 

42 

816 

277 

.0837 

4i>l 

18 

42 

088 

662 

.0464 

297 

18 

43 

837 

331 

.0831 

472 

17 

43 

109 

618 

.0458 

277 

17 

4-1 

859 

385 

.0824 

452 

16 

44 

130 

673 

.0452 

257 

16 

45 

.67880 

.92439 

1.0818 

.7:5432 

15 

45 

.69151 

.95729 

1.0446 

.72236 

15 

4(j 

901 

493 

.0812 

413 

14 

46 

172 

785 

.0440 

216 

14 

47 

923 

547 

.0805 

393 

13 

47 

193 

841 

.0434 

196 

13 

48 

944 

601 

.0799 

373 

12 

48 

214 

897 

.0428 

176 

12 

49 

965 

655 

.0793 

353 

11 

49 

235 

.95952 

.0422 

156 

11 

50 

.67987 

.92709 

1.0786 

.73333 

10 

50 

.69256 

.96008 

1.0416 

.72136 

10 

51 

.68008 

763 

.0780 

314 

9 

51 

277 

064 

.0410 

116 

9 

52 

029 

817 

.0774 

294 

8 

52 

298 

120 

.0404 

095 

8 

53 

051 

872 

.0768 

274 

7 

53 

319 

176 

.0398 

075 

7 

54 

072 

926 

.0761 

254 

6 

64 

340 

232 

.0392 

056 

6 

55 

.68093 

.92980 

1.0755 

.73234 

5 

55 

.69361 

.96288 

1.0385 

.72035 

5 

56 

115 

.93034 

.0749 

215 

4 

56 

382 

344 

.0379 

.72015 

4 

57 

136 

088 

.0742 

195 

3 

57 

403 

400 

.0373 

.71995 

3 

58' 

157 

143 

.0736 

175 

2 

58 

424 

457 

.0367 

974 

2 

59 

179 

197 

.0730 

155 

1 

59 

445 

513 

.0361 

954 

1 

60 

.68200 

.93252 

1.0724 

.731.35 

0 

60 

.69466 

.96569 

1.0355 

.719134 

0 

Cos 

Ctn 

Tan 

Sin 

t 

Cos 

Ctn 

Tan 

Sin   ' 

4.7^ 


J.AO 


44 


44°  —  Values  of  Trigonometric  Functions 


1 

Sin 

Tan 

Ctn 

Cos 

0 

.69466 

.96569 

1.0355 

.71934 

60 

1 

487 

625 

.0349 

914 

59 

2 

508 

681 

.0343 

894 

68 

3 

529 

738 

.0337 

873 

57 

4 

549 

794 

.0331 

853 

56 

6 

.69570 

.96850 

1.0325 

.71833 

55 

6 

591 

907 

.0319 

813 

54 

7 

612 

.96963 

.0313 

792 

53 

8 

633 

.97020 

.0307 

772 

52 

9 

654 

076 

.0301 

752 

51 

10 

.69675 

.97133 

1.0295 

.71732 

50 

11 

696 

189 

.0289 

711 

49 

12 

717 

246 

.0283 

691 

48 

13 

737 

302 

.0277 

671 

47 

14 

768 

359 

.0271 

650 

46 

15 

.69779 

.97416 

1.0265 

.71630 

45 

16 

800 

472 

.0259 

610 

44 

17 

821 

529 

.0253 

590 

43 

18 

842 

586 

.0247 

569 

42 

19 

862 

643 

.0241 

549 

41 

20 

.69883 

.97700 

1.0235 

.71529 

40 

21 

904 

756 

.0230 

508 

39 

22 

925 

813 

.0224 

488 

38 

23 

946 

870 

.0218 

468 

37 

24 

966 

927 

.0212 

447 

36 

25 

.69987 

.97984 

1.0206 

.71427 

35 

26 

.70008 

.98041 

.0200 

407 

34 

27 

029 

098 

.0194 

386 

33 

28 

049 

155 

.0188 

366 

32 

29 

070 

213 

.0182 

345 

31 

30 

.70091 

.98270 

1.0176 

.71325 

30 

31 

112 

327 

.0170 

305 

29 

32 

132 

384 

.0164 

284 

28 

33 

153 

441 

.0158 

264 

27 

34 

174 

499 

.0152 

243 

26 

35 

.70195 

.98556 

1.0147 

.71223 

25 

36 

215 

613 

.0141 

203 

24 

37 

236 

671 

.0135 

182 

23 

38 

257 

728 

.0129 

162 

22 

39 

277 

786 

.0123 

141 

21 

40 

.70298 

.98843 

1.0117 

.71121 

20 

41 

319 

901 

.0111 

100 

19 

42 

339 

.98958 

.0105 

080 

18 

43 

360 

.99016 

.0099 

059 

17 

44 

381 

073 

.0094 

039 

16 

45 

.70401 

.99131 

1.0088 

.71019 

15 

46 

422 

189 

.0082 

.70998 

14 

47 

443 

247 

.0076 

978 

13 

48 

463 

304 

.0070 

957 

12 

49 

484 

362 

.0064 

937 

11 

50 

.70505 

.99420 

1.0058 

.70916 

10 

61 

525 

478 

.0052 

896 

9 

52 

546 

536 

.0047 

875 

8 

63 

567 

594 

.0041 

855 

7 

64 

587 

652 

.0035 

834 

6 

55 

.70608 

.99710 

1.0029 

.70813 

5 

66 

628 

768 

.0023 

793 

4 

67 

649 

826 

.0017 

772 

3 

58 

670 

884 

.0012 

752 

2 

59 

690 

.99942 

.0006 

731 

1 

60 

.70711 

1.0000 

1.0000 

.70711 

0 

Cos 

Ctn 

Tan 

Sin 

f 

45° 


TABLE   III 
COMMON   LOGAEITHMS 

OF    THE 

TEIGONOMETKIC    FUNCTIONS 

FROM 

0°  TO  90°  AT  INTERVALS  OF  ONE  MINUTE 

TO 

FIVE  DECIMAL  PLACES 


TABLE  Ilia  — AUXILIARY  TABLE  OF  S  AND  T  FOR  A  IN  MINUTES 

S  =  log  sin  A  —  log  A'    and     T=  log  tan  A  —  log  A' 


A' 

S  +  10 

0'-  13' 

6.46373 

14'—  42' 

72 

43'—  58' 

71 

59'—  71' 

6.46370 

72'-  81' 

69 

82'—  91' 

68 

92'—  99' 

6.46367 

100'  - 107' 

m 

108'  — 115' 

65 

116'  — 121' 

6.46364 

122'  - 128' 

63 

129'  -  134' 

62 

135'  - 140' 

6.46361 

141'  - 146' 

60 

147'  - 151' 

59 

152'  - 157' 

6.46358 

158'  - 162' 

57 

163' -167' 

56 

168'  -  171' 

6.46355 

172'  - 176' 

54 

177' -181' 

53 

A' 

T  +  10 

0' 

—  26' 

6.46373 

27' 

-  39' 

74 

40' 

-  48' 

75 

49' 

-  56' 

6.46376 

57' 

-  63' 

77 

64' 

-  69' 

78 

70' 

-  74' 

6.46379 

75' 

-  80' 

80 

81' 

-  85' 

81 

86' 

-  89' 

6.46382 

90' 

-  94' 

83 

95' 

—  98' 

84 

99' 

-102' 

6.46385 

103' 

—  106' 

86 

107' 

-110' 

87 

111' 

-113' 

6.46388 

114' 

-117' 

89 

118' 

—  120' 

90 

121' 

— 124' 

6.46391 

125' 

—  127' 

92 

128' 

—  130' 

93 

A' 

r  +  10 

131' 

-133' 

6.46394 

134' 

—  136' 

95 

137' 

—  139' 

96 

140' 

-142' 

6.46397 

143' 

—  145' 

98 

146' 

-148' 

99 

149' 

-150' 

6.46400 

151' 

— 153' 

01 

154' 

- 156' 

02 

157' 

—  158' 

6.46403 

159' 

-  161' 

04 

162' 

— 163' 

05 

164' 

-166' 

6.46406 

167' 

-  168' 

07 

169' 

—  171' 

08 

172' 

-173' 

6.46409 

174' 

-175' 

10 

176' 

-178' 

11 

179' 

—  180' 

6.46412 

181' 

-182' 

13 

183' 

—  184' 

14 

For  small  angles :       log  sin  A  =  log  A'  -}-  S    and    log  tan  A  =  A'  +  T 

For  angles  near  90° :  log  cos  A  =  log  (90°  —  A)' +  S,       log  ctn  A  =  log  (90°—  A)' +T 

where  A'  =  number  of  minutes  in  A ,  and   (90°  —  A)'  =  number  of  minutes  in  90°  —  A 


45 


46 

0° 

—  Logarithms  of  Trigonometric  Functions 

[in 

/ 

LSin 

d 

LTan 

cd 

LCtn 

L  Cos 

0 

0.00  000 
0.00  000 

60 

59 

1 

6.46  373 

30103 
17609 
12494 
9691 

6.46  373 

30103 
17609 
12494 
9691 
7918 
6694 
5800 
5115 
4576 
4139 
3779 
3476 
3219 
2996 

3.53  627 

2 

6.76  476 

6.76  476 

3.23  524 

0.00  000 

58 

3 

6.94  085 

6.94  085 

3.05  915 

0.00  000 

57 

4 

7.06  579 

7.06  579 

2.93  421 

0.00  000 

56 

5 

7.16  270 

7918 

6694- 

5800 

5115 

4576 

4139 

3779 

3476 

3218 

2997 

7.16  270 

2.83  730 

0.00  000 

55 

6 

7.24  188 

7.24  188 

2.75  812 

0.00  000 

54 

CM 

a:) 

•+3 

7 

7.30  882 

7.30  882 

2.69  118 

0.00  000 

53 

o 

8 

7.36  682 

7.36  682 

2.63  318 

0.00  000 

52 

Kfl 

H 

9 
10 

7.41  797 
7.46  373 

7.41  797 
7.46  373 

2.58  203 
2.53  627 

0.00  000 
0.00  000 

51 
50 

4J 

4-3 

CD 

11 
12 

7.50  512 
7.54  291 

7.50  512 
7.54  291 

2.49  488 
2.45  709 

0.00  000 
0.00  000 

49 

48 

1 

CQ 
02 

13 

7.57  767 

7.57  767 

2.42  233 

0.00  000 

47 

io 

r^ 

14 

7.60  985 

7.60  986 

2.39  014 

0.00  000 

46 

p-i 

-* 

r^ 

^ 

Q 

15 

7.63  982 

2802 
2633 
2483 
2348 
2227 

7.63  982 

2803 
2633 
2482 
2348 
2228 

2.36  018 

0.00  000 

45 

o 

^ 

"B 

<D 

16 

7.66  784 

7.66  785 

2.33  215 

0.00  000 

44 

^■"^ 

^ 

CQ 

% 

17 

7.69  417 

7.69  418 

2.30  582 

9.99  999 

43 

w 

CQ 

^ 

18 

7.71  900 

7.71  900 

2.28  100 

9.99  999 

42 

M 

1—* 

P-t 

19 

7.74  248 

7.74  248 

2.25  752 

9.99  999 

41 

ri 

ce 

\—< 

•  l-l 

9 

B 

20 

7.76  475 

2119 
2021 
1930 

1848 
1773 

7.76  476 

2119 
2020 
1931 
1848 
1773 

2.23  524 

9.99  999 

40 

■^ 

<D 

TJ 

U 

a 

21 

7.78  594 

7.78  595 

2.21  405 

9.99  999 

39 

I 

0) 

O 

•i-i 

22 

7.80  615 

7.80  615 

2.19  385 

9.99  999 

38 

rg 

(D 

^ 

23 
24 

7.82  545 
7.84  393 

7.82  546 
7.84  394 

2.17  454 
2.16  606 

9.99  999 
9.99  999 

37 

36 

CD 

0^ 

3 

4.3 

O 

25 

7.86  166 

1704 

7.86  167 

1704 

2.13  833 

9.99  999 

35 

o; 

OQ 

i^ 

OQ 

T^ 

26 

7.87  870 

1639 
1579 
1524 
1472 

7.87  871 

1639 
1579 
1524 
1473 

2.12  129 

9.99  999 

34 

"S) 

^ 

c3 

4-3 

15 

4-= 

CD 

g 

27 

7.89  509 

7.89  510 

2.10  490 

9.99  999 

33 

fl 
c^ 

o 

28 

7.91088 

7.91  089 

2.08  911 

9.99  999 

32 

00 

t>-. 

29 

7.92  612 

7.92  613 

2.07  387 

9.99  998 

31 

tM 
O 

g 

•  1—1 

5 

30 

7.94  084 

1424 

7.94  086 

1424 

2  05  914 

9.99  998 

30 

OQ 

^ 

c^ 

o 

P 

31 

7.95  508 

1379 
1336 
1297 
1259 

7.95  510 

1379 
1336 
1297 
1259 

2.04  490 

9.99  998 

29 

4^ 

CD 

bJO 

^3 

'""' 

C^ 

*^ 

32 
33 

7.96  887 
7  98  223 

7.96  889 
7.98  225 

2.03  111 
2.01  775 

9.99  998 
9.99  998 

28 
27 

u 

<D 
<D 

03 

^ 

o 

<D 

34 

7.99  520 

7.99  522 

2.00  478 

9.99  998 

26 

ci 

35 

8.00  779 

1223 

8.00  781 

1223 

1.99  219 

9.99  998 

25 

"•^ 

o 

T-H 

«+-! 

36 

8.02  002 

1190 
1158 
1128 
1100 

8.02  004 

1190 
1159 
1128 
1100 

1.97  996 

9.99  998 

24 

g 

CQ 

a; 

37 

8.03  192 

8.03  194 

1.96  806 

9.99  997 

23 

^ 

8 

d 

38 

8.04  350 

8.04  353 

1.95  647 

9.99  997 

22 

s 

"Ejo 

1 

=S 

<D 

39 

8.05  478 

8.05  481 

1.94  519 

9.99  997 

21 

7X 

^ 
c^ 

^ 

<D 

40 

8.06  578 

1072 

8.06  581 

1072 

1.93  419 

9.99  997 

20 

m 

^H 

S 

T^ 

41 

8.07  650 

1046 

8.07  653 

1047 

1022 

998 

1.92  347 

9.99  997 

19 

o 

O 

?-i 

^ 

2 

42 

8.08  696 

1022 
999 

8.08  700 

1.91  300 

9.99  997 

18 

OQ 

rO 

M 

43 

8.09  718 

8.09  722 

1.90  278 

9.99  997 

17 

OQ 

a 

^ 

M 

& 

44 

8.10  717 

976 

8.10  720 

976 

1.89  280 

9.99  996 

16 

s, 

il 

<D 
4J 

45 

8.11693 

954 

8.11696 

955 

1.88  304 

9.99  996 

15 

1 

O 

46 

8.12  647 

934 

8.12  651 

934 
915 

895 

878 

1.87  349 

9.99  996 

14 

^ 

r^ 

O 

47 

8.13  581 

914 
896 

877 

8.13  585 

1.86  415 

9.99  99() 

13 

c3 

o 
o 

4-3 

^H 

48 

8.14  495 

8.14  500 

1.85  500 

9.99  996 

12 

o^ 

rt 

cn 

49 

8.15  391 

8.15  395 

1.84  605 

9.99  996 

11 

j_i 

o 

.2 

>* 

50 

8.16  268 

860 

8.16  273 

860 

1.83  727 

9.99  995 

10 

o 

a; 

>■ 

;h 

% 

51 

8.17  128 

843 

8.17  133 

843 

1.82  867 

9.99  995 

9 

ph 

r^ 

o 
o 

u 

52 

8.17  971 

827 

8.17  976 

828 

1.82  024 

9.99  995 

8 

02 

53 

8.18  798 

812 
797 

8.18  804 

812 
797 

1.81 196 

9  99  995 

7 

§ 

u 

54 

8.19  610 

8.19  616 

1.80  384 

9.99  995 

6 

O 

55 

8.20  407 

782 

8.20  413 

782 

1.79  587 

9.99  994 

5 

56 

8.21 189 

769 

8.21 195 

769 

1.78  805 

9.99  994 

4 

57 

8.21  958 

755 

8.21  964 

756 
742 
730 

1.78  036 

9.99  994 

3 

58 

8.22  713 

743 
730 

8.22  720 

1.77  280 

9.99  994 

2 

59 

8.23  456 

8.23  462 

1.76  538 

9,99  994 

1 

60 

8.24  186 

8.24192 

1.75  808 

9.99  993 

0 

LCos 

d 

LCtn 

Cd 

L  Tan 

L  Sin 

/ 

S9°  —  TiOe^ariflims  of  Tricftnoiiifttrip,  FimHions 


Ill] 

1° 

—  Logarithms  of  Trigonometric  Functions 

47 

/ 

LSin  1  d 

LTan 

cd 

LCtn 

LCos 

Prop.  Pts. 

0 

8.24  186 

717 
706 

8.24  192 

718 
706 

1.75  808 

9.99  993 

60 

1 

8.24  ^X)3 

8.24  910 

1.75  0^)0 

9.99  993 

59 

720 

710 

690 

680  670 

2 

8.25  609 

8.25  616 

1.74  384 

9.99  993 

58 

2 

144 

142 

138 

1.36  134 

3 
4 

8.26  304 
8.26  988 

695 
684 
673 
'663 

8.26  312 
8.26  996 

696 
684 
673 
663 

1.73688 
1.73  004 

9.99  993 
9.99  992 

57 
56 

3 
4 
5 

216 

288 
360 

213 

284 
355 

207 
276 
345 

204  201 
272  268 
340  335 

5 

8.27  661 

8.27  669 

1.72  331 

9.99  992 

55 

6 

7 

432 
504 

426 

497 

414 

483 

408  402 

476  469 

(i 

8.28  324 

653 
644 

8.28  332 

654 
643 

1.71  668 

9.99  992 

54 

8 

576 

568 

552 

544  536 

7 

8.28  977 

8.28  986 

1.71  014 

9.99  992 

53 

9 

648 

639 

621 

612  603 

8 

8.29  621 

8.29  629 

1.70  371 

9.99  992 

52 

9 

8.30  255 

634 

8.30  263 

634 

1.69  737 

9.99  991 

51 

660 

650 

640 

630  620 

624 

625 

2 

132 

130 

128 

126  124 

10 

8.;^  879 

616 
608 
599 

8.30  888 

617 
607 
599 

1.69112 

9.99  991 

50 

3 

198 

195 

192 

189  186 

11 
12 

8.31  495 

8.32  103 

8.31  505 

8.32  112 

1.68  495 

1.67  888 

9.99  991 
9  99  9^)0 

49 

48 

4 
5 
6 

264 
330 
396 

260 
325 
390 

256 
320 

384 

252  248 
315  310 
378  372 

13 

8.32  702 

8.32  711 

1.67  289 

9.99  990 

47 

7 

462 

455 

448 

441  434 

14 

8.33  292 

590 
583 

8.33  302 

591 
584 

1.66  698 

9.99  990 

4(5 

8 
9 

528 
594 

520 
585 

512 
576 

504  496 
567  558 

15 

8.33  875 

575 
568 
560 
553 
547 
539 

8.33  886 

575 

568 

1.66  114 

9.99990 

45 

16 

8.34  450 

8.34  461 

1 .65  539 

9  99  989 

44 

610 

600 

590 

580  570 

17 

8.e35  018 

8.35  029 

1.64  971 

9.99  989 

43 

2 

122 

120 

118 

116  114 

18 
19 

8.35  578 

8.36  131 

8.35  590 

8.36  143 

561 
553 
546 
540 

1.64  410 
1.63  857 

9.99  989 
9.99  989 

42 
41 

3 
4 
5 

183 
244 
305 

180 
240 
300 

177 
236 
295 

174  171 
232  228 
290  285 

20 

8.36  678 

8.36  689 

1.63  311 

9.99  988 

40 

6 

7 

366 

427 

360 
420 

354 
413 

348  342 
406  399 

21 

8.37  217 

533 
526 

8.37  229 

533 
527 

1.62  771 

9.99  988 

39 

8 

488 

480 

472 

464  456 

22 

8.37  750 

8.37  762 

1.62  238 

9.99  988 

38 

9 

549 

540 

531 

522  513 

23 

8.38  276 

8.38  289 

1.61711 

9*99  987 

37 

24 

8.38  796 

520 

8.38  809 

520 

1.61191 

9.99  987 

36 

660 

550 

540 

530  520 

514 

514 

2 

112 

110 

108 

106  104 

25 

8.39  310 

508 
502 
496 

8.39  323 

509 
502 
496 

1.60  677 

9.99  987 

35 

3 

168 

165 

162 

159  156 

26 

27 

8.39  818 

8.40  320 

8.39  832 

8.40  334 

1.60168 
1.59  6(^6 

9.99  986 
9.99  986 

34 
33 

4 
5 
6 

224 
280 
336 

220 
275 
330 

216 
270 
324 

212  208 
265  260 
318  312 

28 

8.40  816 

491 

485 

8.40  830 

491 
486 

1.59170 

9.99  986 

32 

7 

392 

385 

378 

371  364 

29 

8.41  307 

8.41  321 

1.58  679 

9.99  985 

31 

8 
9 

448 
504 

440 
495 

432 
486 

424  416 

477  468 

30 

8.41  792 

480 

8.41  807 

480 
475 
470 
464 
460 
455 

1.58193 

9.99  985 

30 

31 

8.42  272 

474 
470 
464 
459 
455 

8.42  287 

1.57  713 

9.99  985 

29 

510 

500 

490 

480  470 

32 

8.42  746 

8.42  762 

1.57  238 

9.99  984 

28 

2 

102 

100 

98 

96   94 

33 
34 

8.43  216 
8.43  680 

8.43  232 
8.43  696 

1.56  768 
1.56  304 

9.99  984 
9.99  984 

27 
26 

3 
4 
5 

153 
204 
255 

150 
200 
250 

i4;r 

196 
245 

144  141 
192  188 
240  235 

35 

8.44  139 

8.44  156 

1.55  844 

9.99  983 

25 

6 

7 

306 
357 

300 
350 

294 
343 

288  282 
336  329 

36 

8.44  594 

450 
445 
441 
436 

8.44  611 

450 
446 
441 
437 

1.55  389 

9.99  983 

24 

8 

408 

400 

392 

384  376 

37 

8.45  044 

8.45  061 

1.54  939 

9.99  983 

23 

9 

459 

450 

441 

432  423 

38 

8.45  489 

8.45  507 

1.54  493 

9.99  982 

22 

39 

8.45  930 

8.45  948 

1.54  052 

9.99  982 

21 

460 

450 

90 
135 

440 

88 
132 

430  420 

86   84 
129   126 

40 

8.46  366 

433 

8.46  385 

432 

1.53  615 

9.99  982 

20 

2 
3 

138 

41 
42 

8.46  799 

8.47  226 

427 
424 

8.46  817 

8.47  245 

428 
424 

1.53  183 
1.52  755 

9.99  981 
9.99  981 

19 

18 

4 
5 

6 

184 
230 
276 

180 
225 
270 

176 
220 
264 

172   168 
215  210 
258  252 

43 

8.47  650 

419 
416 

8.47  689 

420 
416 

1.52  331 

9.99  981 

17 

7 

322 

315 

308 

301  294 

44 

8.48  069 

8.48  089 

1.51911 

9.99  980 

16 

8 
9 

368 
414 

360 
405 

352 
396 

344  336 
387  378 

45 

8.48  485 

411 

8.48  505 

412 

1.51  495 

9.99  980 

15 

4() 

8.48  896 

408 

8.48  917 

408 

1.51083 

9.99  979 

14 

410 

400 

395 

390  385 

47 

8.49  304 

404 
400 

8.49  325 

404 
401 

1.50  675 

9.99  979 

13 

2 

82 

80 

79.0 

78   77.0 

48 

8.49  708 

8.49  729 

1.50  271 

9.99  979 

12 

3 

4 

123 
164 

120  118.5 
160  158.0 

117  115.5 
156  154.0 

49 

8.50  108 

396 

8.50  130 

397 

1.49  870 

9.99  978 

11 

5 

205 

200  197.5 

195  192.5 

50 

8.50  504 

393 

8.50  527 

393 

1.49  473 

9.99  978 

10 

6 

7 

246 

287 

240  237.0 
280  276.5 

234  231.0 
273  269.5 

51 

8.50  897 

390 

8.50  920 

390 

1.49080 

9.99  977 

9 

8 

328 

320  316.0 

312  308.0 

52 

8.51  287 

386 

8.51  310 

386 

1.48  690 

9.99  977 

8 

9 

369 

360  355.5 

351  346.5 

53 
54 

8.51  673 

8.52  055 

382 
379 

8.51  696 

8.52  079 

383 
380 

1.48  304 
1.47  921 

9.99  977 
9.99  976 

7 
6 

2 

380 

76 

375 

75.0 

370 

74 

365  360 

73.0  72 

55 

8.52  434 

376 

8.52  459 

376 

1.47  541 

9.99  976 

5 

3 

114 

112.5 

111 

109.5  108 

56 
57 

8.52  810 

8.53  183 

373 
369 

8.52  835 

8.53  208 

373 
370 

1.47  165 
1.46  792 

9.99  975 
9.99  975 

4 
3 

4 
5 
6 

152 
190 
228 

150.0 
187.5 
225.0 

148 
185 
222 

146.0  144 
182.5  180 
219.0  216 

58 

8.53  552 

367 
363 

8.53  578 

367 
363 

1.46  422 

9.99  974 

2 

7 

266 

262.5 

259 

255.5  252 

59 

8.53  919 

8.53  945 

1.46  055 

9.99  974 

1 

8 
9 

304 
342 

300.0 
337.5 

296 
333 

292.0  288 
328.5  324 

60 

8.54  282 

8.54  308 

1.45  692 

9.99  974 

0 

LGos 

d 

LCtn 

Cd 

LTan 

LSin 

1 

Prop 

.  Pts.      1 

88°  — Logarithms  of  Trigonometric  Functions 


2°  —  Logarithms  of  Trigonometric  Functions 


[in 


r 
0 

LSin 

d 

LTan 

cd 

LCtn 

LGos 

Prop.  Pts. 

8.54  282 

8.54  308 

1.45  692 

9.99  974 

60 

1 

8.54  642 

360 

8.54  669 

361 

1.45  331 

9.99  973 

59 

2 

8.54  999 

357 

8.55  027 

358 

1.44  973 

9.99  973 

58 

3 

8.55  354 

355 

8.55  382 

355 

1.44  618 

9.99  972 

67 

4 

8.55  705 

351 
349 

8.55  734 

352 
349 

1.44  266 

9.99  972 

66 

5 

8.56054 

8.56  083 

1.43  917 

9.99  971 

55 

6 

8.56400 

346 

8.56  429 

346 

1.43  571 

9.99  971 

54 

360      355      350      345 

7 

8.56  743 

343 

8.56  773 

344 

1.43  227 

9.99  970 

63 

2 

72      71.0      70      69.0 

8 

8.57  084 

341 
337 
336 
332 

8.57114 

341 

1.42  886 

9.99  970 

52 

3 

4 

108    106.5    105    103.5 
144    142.0    140    138  0 

9 

8.57  421 

8.57  452 

338 

1.42  548 

9.99  969 

51 

5 

180    177.5    175    172^5 

10 

8.57  757 

8.57  788 

336 

1.42  212 

9.99  969 

50 

6 

7 

216    213.0    210    207.0 
252    248.5    245    241.5 

11 

8.58  089 

8.58  121 

333 

1.41  879 

9.99  968 

49 

8 

288    284.0    280    276.0 

12 

8.58  419 

330 

8.58  451 

330 

1.41  549 

9.99  968 

48 

9 

324    319.5    315    310.5 

13 

8.58  747 

328 

8.58  779 

328 

1.41221 

9.99  967 

47 

14 

8.59  072 

325 

8.59 105 

326 

1.40  895 

9.99  967 

46 

15 

8.59  395 

323 
320 

8.59428 

323 
321 

1.40  572 

9.99  967 

45 

2 

340      335      330      325 

68      67  0      66      ft^  0 

16 

8.59  715 

8.59  749 

1.40  251 

9.99  966 

44 

3 

102    100.5      99      97.5 

17 

8.60  033 

318 

8.60068 

319 

1.39  932 

9.99  966 

43 

4 

136    134.0    132    130.0 

18 

8.60  349 

316 

8.60  384 

316 

1.39616 

9.99  965 

42 

5 
6 

170    167.5    165    162.5 
204    201.0    198    195  0 

19 

8.60  662 

313 

8.60  698 

314 

1.39  302 

9.99  964 

41 

7 

238    234.5    231    227.5 

20 

8.60  973 

311 
309 

8.61  009 

311 

1.38  991 

9.99  964 

40 

8 
9 

272    268.0    264    260.0 
306    301.5    297    292.5 

21 

8.61 282 

8.61  319 

310 

1.38  681 

9.99  963 

39 

22 

8.61 589 

307 

8.61 626 

307 

1.38  374 

9.99  963 

38 

23 

8.61  894 

305 

8.61  931 

305 

1.38  069 

9.99  962 

37 

320      315      310      305 

24 

8.62 196 

302 

8.62234 

303 

1.37  766 

9.99  962 

36 

2 

64      63.0      62      61.0 

25 

8.62  497 

301 
298 

8.62  535 

301 
299 

1.37  465 

9.99  961 

35 

3 

4 

96      94.5      93      91.5 
128    126.0    124    122  0 

26 

8.62  795 

8.62  834 

1.37  166 

9.99  961 

34 

5 

160    157.5    155    152.5 

27 
28 

8.63  091 
8.63  385 

296 
294 
293 
290 

288 

8.63  131 
8.63426 

297 
295 

1.36  869 
1.36  574 

9.99  960 
9.99  960 

33 
32 

6 

7 
8 

192    189.0    186    183.0 
224    220.5    217    213.5 
256    252.0    248    244.0 

29 

8.63  678 

8.63  718 

292 

1.36  282 

9.99  959 

31 

9 

288    283.5    279    274.5 

30 

8.63  968 

8.64009 

291 

1.35  991 

9.99  959 

30 

31 

8.64256 

8.64  298 

289 

1.35  702 

9.99  958 

29 

32 

8.64  543 

287 

8.64  585 

287 

1.35  415 

9.99  958 

28 

300      295       290      285 

33 

8.64  827 

284 
283 

8.64  870 

285 

1.35  130 

9.99  957 

27 

2 
3 

60      59.0      58      57.0 
90       88  5       87       S'l  "i 

34 

8.65  110 

8.65  154 

284 

1.34  846 

9.99  956 

26 

4 

i7\J              00.«_F              <J  §                OtJ.fJ 

120     118.0    116    114.0 

35 

8.65  391 

281 
279 

8.65  435 

281 
280 

1.34  565 

9.99  956 

25 

5 
6 

150    147.5    145    142.5 
180    177.0    174     171.0 

36 

8.65  670 

8.65  715 

1.34285 

9.99  955 

24 

7 

210    206.5    203     199.5 

37 

38 

8.65  947 

8.66  223 

277 
276 

8.65  993 

8.66  269 

278 
276 

1.34  007 
1.33  731 

9.99  955 
9.99  954 

23 
22 

8 
9 

240    236.0    232    228.0 
270    265.5    261    256.5 

39 

8.66  497 

274 
272 
270 
269 

8.66  543 

274 
273 
271 
269 

1.33457 

9.99  954 

21 

40 

8.66  769 

8.66  816 

1.33 184 

9.99  953 

20 

280      275      270      265 

41 

8.67  039 

8.67  087 

1.32  913 

9.99  952 

19 

2 

56      55.0      64      53.0 

42 

8.67  308 

8.67  356 

1.32  644 

9.99  952 

18 

3 

84      82.5      81       79.5 

43 

8.67  575 

267 
266 

8.67  624 

268 
266 

1.32  376 

9.99  951 

17 

4 
5 

112    110.0    108    106.0 
140    137.5    135    132.5 

44 

8.67  841 

8.67  890 

1.32 110 

9.99  951 

16 

6 

168    165.0    162    159.0 

45 

8.68 104 

263 
263 

8.68  154 

264 
263 

1.31  846 

9.99  950 

15 

7 
8 

196    192.5    189    185.5 
224    220.0    216    212.0 

46 

8.68  367 

8.68  417 

1.31  583 

9.99  949 

14 

9 

252    247.5    243    238.5 

47 

8.68  627 

260 

8.68  678 

261 

1.31  322 

9.99  949 

13 

48 

8.68  886 

259 

8.68  938 

260 

1.31062 

9.99  948 

12 

49 

8.69144 

258 
256 
254 
253 

8.69 196 

258 
257 
255 

1.30  804 

9.99  948 

11 

260      255      250      245 

50 

8.69400 

8.69  453 

1.30547 

9.99  947 

10 

2 
3 

52      51.0      50      49.0 
78      76  5      75      73  5 

51 

8.69  654 

8.69  708 

1.30  292 

9.99  946 

9 

4 

104    102.0    100      98!0 

52 

8.69  907 

8.69  962 

254 

1.30  038 

9.99  946 

8 

5 

130    127.5    125    122.5 

53 

8.70 159 

252 
250 

8.70  214 

252 
251 

1.29  786 

9.99  945 

7 

6 

7 

156    153.0    150    147.0 
182    178.5    175    171.5 

54 

8.70409 

8.70  465 

1.29535 

9.99  944 

6 

8 

208    204.0    200    196.0 

55 

8.70  658 

249 

247 

8.70  714 

249 
248 

1.29  286 

9.99  944 

5 

9 

234    229.5    225    220.5 

56 

8.70  905 

8.70  962 

1.29038 

9.99  943 

4 

57 

8.71 151 

246 
244 
243 
242 

8.71  208 

246 
245 
244 
243 

1.28  792 

9.99  942 

3 

58 

8.71  395 

8.71 453 

1.28  547 

9.99  942 

2 

59 

8.71638 

8.71697 

1.28  303 

9.99  941 

1 

60 

8.71 880 

8.71  940 

1.28  060 

9.99  940 

0 

LGos 

d 

LCtn 

Cd 

LTan 

LSin 

/ 

Prop.  Pts. 

87°  —  Logarithms  of  Trigonometric  Functions 


Ill] 


3°  —  Logarithms  of  Trigonometric  Functions 


4S 


L  Sin 


LTan 


cd 


LCtn 


L  Cos 


Prop.  Pts. 


0 

8.71  880 

1 

8.72  120 

2 

8.72  359 

3 

8.72  597 

4 

8.72  834 

5 

8.73  069 

6 

8.73  303 

7 

8.73  535 

8 

8.73  767 

9 

8.73  997 

10 

8.74  226 

11 

8.74  454 

12 

8.74  680 

13 

8.74  906 

14 

8.75  130 

15 

8.75  353 

16 

8.75  575 

17 

8.75  795 

18 

8.76  015 

19 

8.76  234 

20 

8.76  451 

21 

8.76  667 

22 

8.76  883 

23 

8.77  097 

24 

8.77  310 

25 

8.77  522 

26 

8.77  733 

27 

8.77  943 

28 

8.78  152 

29 

8.78  360 

30 

8.78  568 

31 

8.78  774 

32 

8.78  979 

33 

8.79  183 

34 

8.79  386 

35 

8.79  588 

36 

8.79  789 

37 

8.79  990 

38 

8.80  189 

39 

8.80  388 

40 

8.80  585 

41 

8.80  782 

42 

8.80  978 

43 

8.81 173 

44 

8.81  367 

45 

8.81  560 

46 

8.81  752 

47 

8.81  944 

48 

8.82  134 

49 

8.82  324 

50 

8.82  513 

51 

8.82  701 

52 

8.82  888 

53 

8.83  075 

54 

8.83  261 

55 

8.83  446 

56 

8.83  630 

57 

8.83  813 

58 

8.83  996 

59 

8.84 177 

60 

8.84  358 

240 
239 
238 
237 
235 
234 
232 
232 
230 
229 
228 
226 
226 
224 
223 
222 
220 
220 
219 
217 
216 
216 
214 
213 
212 
211 
210 
209 
208 
208 
206 
205 
204 
203 
202 
201 
201 
199 
199 
197 
197 
196 
195 
194 
193 
192 
192 
190 
190 
189 
188 
187 
187 
186 
185 
184 
183 
183 
181 
181 


8.71  940 

8.72  181 
8.72  420 
8.72  659 

8.72  896 
8.73 132 

8.73  366 
8.73  600 

8.73  832 
8.74063 

8.74  292 
8.74  521 
8.74  748 

8.74  974 

8.75  199 
8.75  423 
8.75  645 

8.75  867 

8.76  087 
8.76  306 
8.76  525 
8.76  742 

8.76  958 

8.77  173 
8.77  387 
8.77  600 

8.77  811 

8.78  022 
8.78  232 
8.78  441 
8.78  649 

8.78  855 
8.79061 

8.79  266 
8.79  470 
8.79  673 

8.79  875 

8.80  076 
8.80  277 
8.80  476 
8.80  674 

8.80  872 
8.81 068 

8.81  264 
8.81  459 
8.81  653 

8.81  846 

8.82  038 
8.82  230 
8.82  420 
8.82  610 
8.82  799 

8.82  987 

8.83  175 
8.83  361 
8.83  547 
8.83  732 

8.83  916 
8.84 100 

8.84  282 
8.84  464 


241 
239 
239 
237 
236 
234 
234 
232 
231 
229 
229 
227 
226 
225 
224 
222 
222 
220 
219 
219 
217 
216 
215 
214 
213 
211 
211 
210 
209 
208 
206 
206 
205 
204 
203 
202 
201 
201 
199 
198 
198 
196 
196 
195 
194 
193 
192 
192 
190 
190 
189 
188 
188 
186 
186 
185 
184 
184 
182 
182 


1.28  060 
1.27  819 
1.27  580 
1.27  341 
1.27  104 
1.26  868 
1.26  634 
1.26  400 
1.26168 
1.25  937 
1.25  708 
1.25  479 
1.25  252 
1.25  026 
1.24  801 
1.24  577 
1.24  355 
1.24  133 
1.23  913 
1.23  694 
1.23  475 
1.23  258 
1.23042 
1.22  827 
1.22  613 
1.22  400 
1.22  189 
1.21978 
1.21768 
1.21559 
1.21351 
1.21 145 
1.20  939 
1.20  734 
1.20  530 
1.20  327 
1.20  125 
1.19  924 
1.19  723 
1.19  524 
1.19326 
1.19  128 
1.18  932 
1.18  736 
1.18  541 
1.18  347 
1.18  154 
1.17  962 
1.17  770 
1.17  580 
1.17  390 
1.17  201 
1.17  013 
1.16  825 
1.16  639 
1.16  453 
1.16  268 
1.16  084 
1.15  900 
1.15  718 
1.15  536 


9.99  940 
9.99  940 
9.99  939 
9.99  938 
9.99  938 
9.99937 
9.99  936 
9.99  936 
9.99  935 
9.99  934 
9.99  934 
9.99  933 
9.99  932 
9.99  932 
9.99  931 
9.99  930 
9.99  929 
9.99  929 
9.99  928 
9.99  927 
9.99  926 
9.99  926 
9.99  925 
9.99  924 
9.99  923 
9.99  923 
9.99  922 
9.99  921 
9.99  920 
9.99  920 
9.99  919 
9.99  918 
9.99  917 
9.99  917 
9.99  916 
9.99  915 
9.99  914 
9.99  913 
9.99  913 
9.99  912 
9.99  911 
9.99  910 
9.99  909 
9.99  909 
9.99  908 
9.99  907 
9.99  906 
9.99  905 
9.99  904 
9.99  904 
9.99  903 
9.99  902 
9.99  901 
9.99  900 
9.99  899 
9.99  898 
9.99  898 
9.99  897 
9.99  896 
9  99  895 
9.9^)894 


235 

47.0 
70.5 
94.0 


241  239  237 

48.2  47.8  47.4 

72.3  71.7  71.1 

96.4  95.6  94.8     „..„ 

120.5  119.5  118.5  117.5 

144.6  143.4  142.2   141.0 

168.7  167.3  165.9   164.5 

192.8  191.2  189.6   188.0 

216.9  215.1  213.3  211.5 

234  232  229       227 

46.8     46.4  45.8     45.4 

70.2     69.6  68.7     68.1 

93.6     92.8  91.6     90.8 

117.0  116.0  114.5   113.5 

140.4  139.2  137.4   136.2 

163.8  162.4  160.3   158.9 

187.2  185.6  183.2   181.6 

210.6  208.8  206.1  204.3 


226 

45.2 
67.8 
90.4 
113.0 
135.6 
158.2 
180.8 
203.4 

219 

43.8 
65.7 
87.6 
109.5 
131.4 
153.3 
175.2 
197.1 

211 

42.2 
63.3 

84.4 
105.5 
126.6 
147.7 
168.8 
189.9 


224 


44.8 
67.2 


222 
44.4 


220 

44.0 
66.0 
-_-_  88.0 
112.0  111.0  110.0 
134.4  133.2  132.0 
156.8  155.4  154.0 
179.2  177.6  176.0 
201.6  199.8  198.0 


217 

43.4 
65.1 
86.8 
108.5 
130.2 
151.9 
173.6 
195.3 

208 

41.6 
62.4 
83.2 
104.0 
124.8 
145.6 
166.4 
187.2 


215 

43.0 
64.5 
86.0 
107.5 
129.0 
150.5 
172.0 
193.5 

206 

41.2 
61.8 
82.4 
103.0 
123.6 
144.2 
164.8 
185.4 


213 

42.6 
63.9 
85.2 
106.5 
127.8 
149.1 
170.4 
191.7 

203 

40.6 
60.9 
81.2 
101.5 
121.8 
142.1 
162.4 
182.7 


I 


201 

199 

197 

195 

2 

40.2 

39.8 

39.4 

39.0 

3 

60.3 

59.7 

59.1 

58.5 

4 

80.4 

79.6 

78.8 

78.0 

5 

100.5 

99.5 

98,5 

97.5 

6 

120.6 

119.4 

118.2 

117.0 

V 

140.7 

139.3 

137.9 

136.5 

8 

160.8 

159.2 

157.6 

156.0 

9 

180.9 

179.1 

177.3 

175.5 

193 

192 

190 

188 

2 

38.6 

38.4 

38.0 

37.6 

3 

57.9 

57.6 

57.0 

56.4 

4 

77.2 

76.8 

76.0 

75.2 

5 

96.5 

96.0 

95.0 

94.0 

6 

115.8 

115.2 

114.0 

112.8 

7 

135.1 

134.4 

133.0 

131.6 

s 

154.4 

153.6 

152.0 

150.4 

9 

173.7 

172.8 

171.0 

169.2 

186  184  182  181 

37.2  36.8  36.4  36.2 

55.8  55.2  54.6  54.3 

74.4  73.6  72.8  72.4 

93.0  92.0  91.0  90.5 

111.6  110.4  109.2  108.6 

130.2  128.8  127.4  126.7 

148.8  147.2  145.6  144.8 

167.4  165.6  163.8  162.9 


L  Cos 


LCtn 


cd 


LTan 


L  Sin 


Prop.  Pts. 


86°— Logarithms  of  Trigonometric  Functions 


50 


4°— Logarithms  of  Trigonometric  Functions  [in 


/ 

L  Sin 

d 

LTan 

cd 

LCtn 

LCos 

Prop.  Pts. 

0 

8.84  358 

181 

8.84  464 

1.15  536 

9.99  894 

60 

1 

8.84  539 

8.84  646 

182 

1.15  354 

9.99  893 

59 

182      181      180      179 

2 

8.84  718 

179 

8.84  826 

180 

1.15  174 

9.99  892 

58 

2       36.4     36.2     36.0     35.8 

3 

8.84  897 

179 

178 

8.85  006 

180 

1.14  994 

9.99  891 

57 

3  54.6     54.3     54.0     53.7 

4  72  8     72  4     72  0     71  fi 

4 

8.85  075 

8.85  185 

179 

1.14  815 

9.99  891 

56 

5       91.0     90.5     90.0     89.5 

5 

8.85  252 

177 
177 

8.85  363 

178 
177 

1.14  637 

9.99  890 

55 

6  109.2   108.6  108.0   107.4 

7  127.4  126.7   126.0   125.3 

6 

8.85  429 

8.85  540 

1.14  460 

9.99  889 

54 

8     145.6   144.8   144.0   143  2 

7 

8.85  605 

176 

8.85  717 

177 

1.14  283 

9.99  888 

53 

9     163.8  162.9   162.0   161.1 

8 

8  85  780 

175 

8.85  893 

176 

1.14107 

9.99  887 

62 

9 

8.85  955 

175 
173 

8.86  069 

176 
174 

1.13  931 

9.99  886 

51 

178      177       176       175 
2       35.6     35.4     35.2     35.0 

10 

8.86  128 

8.86  243 

1.13  757 

9.99  885 

50 

3       53.4     53.1      52.8     52.5 

11 

8.86  301 

173 
173 
171 

8.86  417 

174 
174 

1.13  583 

9.99  884 

49 

4  71.2     70.8     70.4     70.0 

5  89.0     88.5     88.0     87.5 

6  106.8  106.2   105.6   105.0 

12 

8.86  474 

8.86  591 

1.13  409 

9.99  883 

48 

13 

8.86  645 

8.8()  763 

172 

1.13  237 

9.99  882 

47 

7     124.6   123.9   123.2   122.5 

14 

8.86  816 

171 
171 

8.86  935 

172 

171 

1.13  065 

9.99  881 

46 

8  142.4   141.6   140.8   140.0 

9  160.2   159.3   158.4   157.5 

15 

8.86  987 

169 
169 

8.87  106 

1.12  894 

9.99  880 

45 

16 

8.87  156 

8.87  277 

171 

1.12  723 

9.99  879 

44 

174      173      172       171 

17 

8.87  325 

8.87  447 

170 

1.12  553 

9.99  879 

43 

2       34.8     .34.6     34.4     34.2 

18 
19 

8.87  494 
8.87  661 

169 
167 

8.87  616 
8.87  785 

169 
169 

1.12  384 
1.12  215 

9.99  878 
9.99  877 

42 

41 

3  52.2     51.9     51.6     51.3 

4  69.6     69.2     68.8     68.4 

5  87.0     86.5     86.0     85.5 

20 

8.87  829 

168 
166 

8.87  953 

168 

1.12  047 

9.99  876 

40 

6  104.4   103.8   103.2   102.6 

7  121.8   121.1    120.4   119.7 

21 

8.87  995 

8.88  120 

167 

1.11  880 

9.99  875 

39 

S     139,2   138.4   137.6   136.8 

22 

8.88  161 

166 
165 
164 
164 

8.88  287 

167 

1.11713 

9.99  874 

38 

9     156.6  155.7  154.8  153.9 

23 
24 

8.88  326 
8.88  490 

8.88  453 
8.88  618 

166 
165 
165 

1.11547 
1.11  382 

9.99  873 
9.99  872 

37 
36 

170      169       168      167 

2       34.0     33.8     33.6     33.4 

25 

8.88  654 

8.88  783 

1.11217 

9.99  871 

35 

3       51.0     50.7     50.4     50.1 

26 

8.88  817 

163 
163 
162 
162 
160 

8.88  948 

165 
163 
16S 
163 
161 

1.11052 

9.99  870 

34 

t       68.0     67.6     67.2     66.8 
5       85.0     84  5     84  0     83  5 

27 

8.88  980 

8.89  111 

1.10  889 

9.99  869 

33 

3     102.0   101.4   100.8   100.2 

28 
29 

8.89  142 
8.89  304 

8.89  274 
8.89  437 

1.10  726 
1.10  563 

9.99  868 
9.99  867 

QO     7    '119.0  118.3   117.6   116.9 
q7     8     136.0   135.2   134.4   133.6 
31     9     153.0   152.1   151.2   150.3 

30 

8.89  464 

161 
159 

8.89  598 

1.10402 

9.99  866 

30 

31 

8.89  625 

8.89  760 

162 

1.10  240 

9.99  865 

29 

166       165       164      163 

32 

8.89  784 

8.89  920 

160 

1.10080 

9.99  864 

9«     2       33.2     33.0     32.8     32.6 

33 

8.89  943 

159 
159 

8.m  080 

160 
160 

1.09  920 

9.99  863 

o7     3       49.8     49.5     49.2     48.9 
^'      4       66.4     66.0     65.6     65.2 

34 

8.90  102 

8.90  240 

1.09  760 

9.99  862 

26     5       83.0     82.5     82.0     81.5 

35 

8.90  260 

158 
157 

8.90  399 

159 
158 

1.09601 

9.99  861 

„c     6       99.6     99.0     98.4     97.8 
25     7     116.2   115.5   114.8  114.1 

36 

8.90  417 

8.90  557 

1.09  443 

9.99  860 

24    ^ 

I     132.8   132.0   131.2   130.4 

37 

8.90574 

157 

8.90715 

158 

1.09  285 

9.99  859 

23    ' 

)  1  149.4  148.5  147.6  146.7 

38 

8.90  730 

156 

8.90  872 

157 

1.09128 

9.99  858 

22 

162       161      160       159 
'       32.4     32.2     32.0     31.8 

39 

8.90  885 

155 
155 

8.91 029 

157 
156 

1.08  971 

9.99  857 

21    . 

40 

8.91 040 

8.91 185 

1.08  815' 

9.99  856 

20    2 

48.6     48.3     48.0     47.7 

41 

8.91 195 

155 
154 

8.91  340 

155 
155 

1.08  660 

9.99855 

19    i 

64.8     64.4     64.0     63.6 
81.0     80  5     80  0     79  5 

42 

8.91  349 

8  91  495 

1.08  505 

9.99  854 

18    e 

97.2     96.6     96.0     95.4 

43 
44 

8.91 502 
8.91655 

153 
153 
152 

8.91  650 
8.91  803 

155 
153 
154 

1.08  350 
1.08  197 

9.99  853 
9.99  852 

17     8 
16    g 

113.4   112.7   112.0   111.3 

129.6   128.8   128.0   1272 

1  145.8   144.9   144.0  143.1 

45 

8.91 807 

152 

8.91  957 

153 

152 

1.08  043 

9.99  851 

15 

46 

8.91 959 

8.92  110 

1.07  890 

9.99  850 

14 

158       157       156       155 

47 

8.92 110 

151 

8.92  262 

1.07  738 

9.99  848 

13     2 

f     31.6      31.4     31.2     31.0 

48 

8.92  261 

151 

8.92  414 

152 

1.07  686 

9.99  847 

12    3 

47.4      47.1      46.8     46.5 

49 

8.92  411 

150 
150 

8.92  565 

151 
151 

1.07  435 

9.99  846 

11    t 

63.2     62.8     62.4     62.0 
79.0     78.5     78.0     77.5 

50 

51 

8.92  561 
8.92  710 

149 
149 

8.92  716 
8.92  866 

150 
150 

1.07  284 
1.07134 

9.99  845 
9.99  844 

10     6 

9     8 

94.8     94.2     93.6     93.0 
110.6   109.9   109.2   108.5 
126.4  125.6   124.8  124.0 

52 

8.92  859 

8.93  016 

1.06  984 

9.99  843 

8    9 

142.2   141.3   140.4  139.5 

53 

8.93  007 

148 

8.93  165 

149 

1.06  835 

9.99  842 

7 

54 

8-93  154 

147 

8.93  313 

148 

1.06  687 

9.99  841 

6 

154      153       152       151 

55 

8.93  301 

147 
147 

8.93  462 

149 
147 

1.06  538 

9.99840 

5  i 

30.8     30.6     30.4     30.2 
46.2     45.9     45.6     45.3 

56 

8.93  448 

8.93  609 

1.06  391 

9.99  839 

4    4 

61.6     61.2     60.8     60.4 

57 

8.93  594 

146 
146 

8.93  756 

147 
147 

1.06  244 

9.99  838 

3    1 

77.0     76.5     76.0     75.5 
92  4     91  8     91.2     90  6 

58 

8.93  740 

8.93  903 

1.06  097 

9.99  837 

2     7 

107.8  107.1    106.4   105.7 

59 

8.93  885 

145 
145 

8.94  049 

146 
146 

1.05  951 

9.99  836 

« « 

123.2   122.4   121.6   120.8 
138.6   137.7   136.8  135.9 

60 

8.94  030 

8.94  195 

1.05  805 

9.99  834 

0 

LCos 

d 

LGtn 

Cd 

L  Tan 

L  Sin 

'    1             Prop.  Pts.            1 

85°  —  Logarithms  of  Trigonometric  Functions 


Ill] 

6° 

—  Logarithms  of  Trigonometric 

Functions           51 

/ 

L  Sin 

d 

LTan 

cd 

LCtn 

LCos 

Prop.  Pts.            1 

0 

8.94  030 

8.94  195 

1.05  805 

9.9V)  834 

60 

1 

8.94174 

144 

8.94  340 

145 

1.05  660 

9.f^  833 

59 

150 

149       148       147 

2 

8.94  317 

143 
144 

8.94  485 

145 
145 

1.05  515 

9.99  832 

58 

2 
3 

30.0 
45.0 

29.8     29.6     29.4 
44  7     44  4     44  1 

3 

8.94  461 

8.94  630 

1.05  370 

9.99  831 

57 

4 

60.0 

59.6     59!2     58^8 

4 

8.94  603 

142 
143 

8.94  773 

143 
144 

1.05  227 

9.99  830 

56 

5 

6 

75.0 
90.0 

74.5     74.0     73.5 
89.4     88.8     88.2 

5 

8.94  746 

8.94  917 

1.05  083 

9.99  829 

55 

7 

105.0 

104.3   103.6   102.9 

6 

7 

8.94  887 

8.95  029 

141 
142 

8.95  060 
8.95  202 

143 
142 

1.04  940 
1.04  798 

9.99  828 
9.()9  827 

54 
53 

8 
9 

120.0 
135.0 

119.2   118.4   117.6 
134.1   133.2   132.3 

8 

8.95  170 

141 

8.95  344 

142 
142 
141 

1.04  656 

9.99  825 

52 

14fi 

145       144       14? 

9 

8.95  310 

140 
140 

8.95  486 

1.04  514 

9.99  824 

51 

2 

29.2 

^^v             ^"ZY             XikO 

29.0     28.8     28.6 

10 

8.95  450 

8.95  627 

1.04  373 

9.99  823 

50 

3 

43.8 

'43.5     43.2     42.9 

11 

8.95  589 

139 

8.95  767 

140 
141 

1.04  233 

9.99  822 

49 

4 
5 

58.4 
73.0 

58.0     57.6     57  2 
72.5     72.0     71  5 

12 

8.95  728 

139 

8.95  908 

1.04  092 

9.99  821 

48 

6 

87.6 

87.0     86.4     85.8 

13 
14 

8.95  867 

8.96  005 

139 
138 
138 

8.96  047 
SA)6  187 

139 
140 
138 

1.03  953 
1.03  813 

9.99  820 
9.99  819 

47 

46 

7 
8 
9 

102.2 
116.8 
131.4 

101.5   100.8   100.1 
116.0  115.2   114.4 
130.5  129.6  128.7 

15 

8.96  143 

8.96  325 

1.03  675 

9.99  817 

45 

16 

8.96  280 

137 

8.96  464 

139 

1.03  536 

9.99  816 

44 

142 

141      140      139 

17 
18 

S/Mi  417 
8.iX>553 

137 

136 

8.96  602 
8.96  739 

138 
137 
138 

1.03  398 
1.03  261 

9.99  815 
9.99  814 

43 
42 

2 
3 

4 

28.4 
42.6 
56.8 

28.2  28.0     27.8 

42.3  42.0     41.7 

56.4  56.0     55.6 

19 

8.9(5  689 

136 

8.96  877 

1.03  123 

9.99  813 

41 

5 

71.0 

70.5     70.0     69.5 

20 

8.96  825 

136 

8.97  013 

136 

1.02  987 

9.99  812 

40 

6 

7 

85.2 
99.4 

84.6  84.0     83.4 

98.7  98.0     97.3 

21 

8.<)6  960 

135 

8.97  150 

137 

1.02  850 

9.99  810 

39 

8 

113.6 

112.8   112.0  111.2 

22 

8.97  095 

135 

8.97  285 

135 

1.02  715 

9.99  809 

38 

9 

127.8 

126.9  126.0  125.1 

23 
24 

8.97  229 
8.97  363 

134 
134 
133 

8.97  421 
8.97  556 

136 
135 
135 

1.02  579 
1.02  444 

9.99  808 
9.99  807 

37 
36 

2 

138 

27.6 

137       136       135 

27.4     27.2     27.0 

25 

8.97  496 

8.97  691 

1.02  309 

9.99  806 

35 

3 

41.4 

41.1     40.8     40.5 

26 

8.97  629 

133 
133 

8.97  825 

134 
134 

1.02  175 

9.99  804 

34 

4 
5 

55.2 
69.0 

54.8     54.4     54.0 
68.5     68.0     67  5 

27 

8.97  762 

8.97  959 

1.02041 

9.99  803 

33 

6 

82.8 

82.2     81.6     81.0 

28 
29 

8.97  894 
8.98026 

132 
132 
131 

8.98092 
8.98  225 

133 
133 
133 

1.01908 
1.01  775 

9.99  802 
9.99  801 

32 
31 

7 
8 
9 

96.6 
110.4 
124.2 

95.9     95.2     94.5 
109.6   108.8   108.0 
123.3   122.4   121.5 

30 

8.98  157 

8.98  358 

132 
132 

1.01  642 

9.99  800 

30 

31 

8,98  288 

131 

8.98  4^)0 

1.01510 

9.99  798 

29 

134 

133       132       131 

32 

8.98  419 

131 

8.98  622 

1.01  378 

9.99  797 

28 

2 

26.8 

26.6     26.4     26.2 

33 

8.98  549 

130 

8.98  753 

131 
131 

1.01  247 

9.99  796 

27 

3 

4 

40.2 
53.6 

39.9     39.6     39.3 
53.2     52  8     52  4 

34 

8.98  679 

130 

8.98  884 

1.01116 

9.99  795 

26 

5 

67.0 

66.5     66.0     65.5 

35 

8.98  808 

129 

8.99  015 

131 
130 

1.00  985 

9.99  793 

25 

6 

7 

80.4 
93.8 

79.8     79.2     78.6 
93.1     92.4     91.7 

36 

8.98  937 

129 

8.99  145 

1.00  855 

9S}9  792 

24 

8 

107.2 

106.4   105.6   104.8 

37 

8.99  066 

129 

8.99  275 

130 

1.00  725 

9.99  791 

23 

9 

120.6 

119.7  118.8  117.9 

38 

8.99194 

128 

8.99  405 

130 

1.00  595 

9.99  790 

22 

,-,« 

129       128      127 
25.8     25.6     25.4 

39 

8.99  322 

128 
128 

8.99  534 

129 
128 

1.00  466 

9.99  788 

21 

2 

26.0 

40 

8.99  450 

8.99  662 

1.00  338 

9.99  787 

20 

3 

39.0 

38.7     38.4     38.1 

41 
42 

8.99  577 
8.99  704 

127 
127 

8.99  791 
8.99  919 

129 
128 

1.00  209 
1.00  081 

9.99  786 
9.99  785 

19 
18 

4 
5 
6 

52.0 
65.0 
78.0 

51.6     51.2     50.8 
64.5     64.0     63.5 
77.4     76.8     76.2 

43 

8.99  830 

126 

9.00  046 

127 

0.99  954 

9.99  783 

17 

7 

91.0 

90.3     89.6     88.9 

44 

8.99  956 

126 
126 

9.00  174 

128 
127 

0.99  826 

9.99  782 

16 

8 
9 

104.0 
117.0 

103.2   102.4   101.6 
116.1   115.2   114.3 

45 

9.00  082 

9.00  301 

126 

0.99  699 

9.99  781 

15 

46 

9.00  207 

125 

9.00  427 

0.99  573 

9.99  780 

14 

126 

125      124      123 

47 

9.00  332 

125 

9.00  553 

126 

0.99  447 

9.99  778 

13 

2 

25.2 

25.0     24.8     24.6 

48 

9.00  456 

124 
125 

9.00  679 

126 
126 

0.99  321 

9.99  777 

12 

3 
4 

37.8 
50  4 

37.5     37.2     36.9 
50.0     49.6     49.2 

49 

9.00  581 

9.00  805 

0.99  195 

9.99  776 

11 

5 

63.0 

62.5     62.0     61.5 

50 

9.00  704 

123 
124 

9.00  930 

125 
125 

0.99070 

9.99  775 

10 

6 

7 

75.6 
88.2 

75.0     74.4     73.8 
87.5     86.8     86.1 

51 

9.00  828 

9.01 055 

0.98  945 

9.99  773 

9 

8 

100.8 

100.0     99.2     98.4 

52 

9.00  951 

123 

9.01 179 

124 

0.98  821 

9.99  772 

8 

9 

113.4 

112.5  111.6   110.7 

53 

9.01074 

123 

9.01  303 

124 

0.98  697 

9.99  771 

7 

54 

9.01 196 

122 

9.01  427 

124 

0.98  573 

9.99  769 

6 

Izz       izx      xzu        1 

122 

123 

2 

24.4      24.2      24.0 

55 

9.01  318 

9.01  550 

0.98  450 

9.99  768 

5 

3 

36.6     36.3     36.0 

56 

9.01  440 

122 

9.01  673 

123 
123 
122 

0.98  327 

9.99  767 

4 

4 
5 

6 

48.8     48.4     48.0 
61.0     60.5     60.0 
73.2      72.6     72.0 

57 

9.01 561 

121 
121 

9.01  796 

0.98  204 

9.99  765 

3 

58 

9.01  682 

9.01  918 

0.98  082 

9.99  764 

2 

7 

85.4     84.7     84.0 

59 

9.01  803 

121 
120 

9.02  040 

122 
122 

0.97  960 

9.99  763 

1 

8 
9 

97.6     96.8     96.0 
109.8  108.9  108.0 

60 

9.01  923 

9.02  162 

0.97  838 

9.99  761 

0 

LCos 

d 

LCtn 

Cd 

LTan 

LSin 

' 

Prop.  Pts. 

84°— Logarithms  of  Trigonometric  Functions 


62 

6° 

—  Logarithms  of  Trigonometric  Functions 

[III 

/ 

LSin 

d 

LTan 

cd 

LCtn 

LCos 

Prop.  Pts.            1 

0 

9.01  923 

9.02  162 

0.97  838 

9.99761 

60 

1 

9.02  043 

120 

9.02  283 

121 

0.97  717 

9.99  760 

69 

2 

9.02  163 

120 

9.02  404 

121 

0.97  596 

9.99  759 

68 

3 

9.02  283 

120 

9.02  525 

121 

0.97  475 

9.99  757 

67 

4 

9.02  402 

119 
118 
119 
118 

9.02  645 

120 
121 
119 
120 

0.97  355 

9.99  756 

66 

5 

9.02  520 

9.02  766 

0.97  234 

9.99  756 

55 

121 

120      119 

118 

6 

9.02  639 

9.02  885 

0.97  115 

9.99  753 

64 

2 

24  2 

24  0     23  8 

23  6 

7 

9.02  757 

9.03  005 

0.96  995 

9.99  752 

53 

3 

36.3 

36.0     35.7 

35'a 

8 
9 

9.02  874 
9.02  992 

117 
118 
117 
117 

9.03  124 
9.03  242 

119 
118 

0.96  876 
0.96  758 

9.99  751 
9.99  749 

62 
51 

4 
5 
6 

48.4 
60.5 
72.6 

48.0     47.6 
60.0     59.5 
72.0     71.4 

47.2 
59.0 

70.8 

10 

9.03 109 

9.03  361 

119 
118 

0.96  639 

9.99  748 

50 

7 
8 

84.7 
96.8 

84.0     83.3 
96  0     95  2 

82.6 
94  4 

11 

9.03  226 

9.03  479 

0.96  521 

9.99  747 

49 

9 

108.9 

108.0  107.1 

10612 

12 

9.03  342 

116 

9.03  597 

118 

0.96  403 

9.99  745 

48 

13 

9.03458 

116 

9.03  714 

117 

0.96  286 

9.99  744 

47 

117 

116       116 

114 

14 

9.03  574 

116 
116 
115 

9.03  832 

118 
116 
117 

0.96  168 

9.99  742 

46 

2 

23.4 

23.2     23.0 

22.8 

15 

16 

9.03690 
9.03  805 

9.03  948 

9.04  065 

0.96  052 
0.95  935 

9.99  741 
9.99  740 

45 

44 

3 
4 
5 

35.1 

46.8 
58.5 

34.8     34.5 
46.4     46.0 
58.0     57.5 

34.2 
45.6 
57.0 

17 

9.03  920 

115 

9.04181 

116 

0.95  819 

9.99  738 

43 

6 

70.2 

69.6     69.0 

68.4 

18 

9.04034 

114 
115 
113 
114 
114 

9.04  297 

116 

0.95  703 

9.99  737 

42 

7 
8 

81.9 
93.6 

81.2     80.5 
92  8     92  0 

79.8 
91  2 

19 

9.04 149 

9.04413 

116 

0.95  587 

9.99  736 

41 

9 

105.3 

104.4  103.5 

102.6 

20 

9.04  262 

9.04  528 

115 

0.95  472 

9.99  734 

40 

21 

9.04  376 

9.04  643 

115 

0.95  357 

9.99  733 

39 

113 

112       111 

110 

22 

9.04  490 

9.04  758 

115 

0.95  242 

9.99  731 

38 

2 

22.6 

22.4     22.2 

22.0 

23 

9.04  603 

113 
112 

9.04  873 

115 

0.95  127 

9.99  730 

37 

3 

4 

33.9 
45  2 

33.6     33.3 
44.8     44  4 

33.0 
44  0 

24 

9.04  715 

9.04  987 

114 

0.95  013 

9.99  728 

36 

5 

56!5 

56^0     55.5 

55^0 

25 

9.04  828 

113 
112 

9.05  101 

114 

0.94  899 

9.99  727 

35 

6 

7 

67.8 
79.1 

67.2     66.6 

78.4     77.7 

66.0 
77.0 

26 

9.04  940 

9.05  214 

113 

0.94  786 

9.99726 

34 

8 

90.4 

89.6     88.8 

88.0 

27 

9.05  052 

112 
112 

9.05  328 

114 

0.94  672 

9.99  724 

33 

9 

101.7 

100.8     99.9 

99.0 

28 

9.05 164 

9.05  441 

113 

0.94  659 

9.99  723 

32 

29 

9.05  275 

111 

9.05  553 

112 

0.94  447 

9.99  721 

31 

109 

108      107 

106 

30 

9.05  386 

111 
111 

9.05  666 

113 

112 

0.94  334 

9.99  720 

30 

2 
3 

21.8 
32.7 

21.6     21.4 
32.4     32.1 

21.2 
31  8 

31 

9.05  497 

9.05  778 

0.94  222 

9.99  718 

29 

4 

43.6 

43.2     42.8 

42.4 

32 
33 

9.05  607 
9.05  717 

110 
110 
110 

9.05  890 

9.06  002 

112 
112 

0.94  110 
0.93  998 

9.99  717 
9.99  716 

28 

27 

5 

6 

7 

54.5 
65.4 
76.3 

64.0     53.5 
64.8     64.2 
75.6     74.9 

53.0 
63.6 
74.2 

34 

9.05  827 

9.06  113 

111 

0.93  887 

9.99  714 

26 

8 

87.2 

86.4     85.6 

84.8 

110 

111 

9 

Q8.1 

97.2     96.3 

95  4 

35 

9.05  937 

109 
109 

9.06  224 

0.93  776 

9.99  713 

25 

36 

9.06  046 

9.06  335 

111 

0.93665 

9.99  711 

24 

37 

9.06 155 

9.06  445 

110 

0.93  655 

9.99  710 

23 

38 

9.06  264 

109 

9.06  556 

111 

0.93  444 

9.99  708 

22 

39 

9.06  372 

108 
109 

9.06  666 

110 
109 

0.93  334 

9.99  707 

21 

40 

9.06  481 

9.06  775 

0.93  225 

9.99  705 

20 

41 

9.06  589 

108 

9.06  885 

110 

0.93 115 

9.99  704 

19 

From  the  top: 

42 
43 

9.06  696 
9.06  804 

107 
108 

9.06  994 

9.07  103 

109 
109 

0.93  006 
0.92  897 

9.99  702 
9.99  701 

18 
17 

For    6°+   or   186°+, 

44 

9.06  911 

107 
107 
106 
107 

9.07  211 

108 

0.92  789 

9.99699 

16 

read  as 

printed ; 

for 

45 

9.07  018 

9.07  320 

109 

0.92  680 

9.99  698 

15 

96°+    or   276°+, 

read 

46 

9.07  124 

9.07  428 

108 

0.92  572 

9.99  696 

14 

co-function. 

47 

9.07  231 

9.07  536 

108 

0.92  464 

9.99  695 

13 

48 

9.07  337 

106 

9.07  643 

107 

0.92  357 

9.99  693 

12 

49 

9.07  442 

105 
106 
105 
105 

9.07  751 

108 
107 
106 
107 

0.92  249 

9.99692 

11 

From  trie  oottom 

•* 

50 

9.07  548 

9.07  858 

0.92  142 

9.99690 

10 

For  83°+  or  263°+.  1 

51 
52 

9.07  653 
9.07  758 

9.07  964 

9.08  071 

0.92  036 
0.91  929 

9.99  689 
9.99  687 

9 

8 

read    as 

printed ; 

for 

53 

9.07  863 

105 

9.08  177 

106 

0.91  823 

9.99  686 

7 

173°+  or  353"+, 

read 

64 

9.07  968 

105 
104 
104 

9.08  283 

106 
106 

0.91  717 

9.99684 

6 

co-function. 

55 

9.08  072 

9.08  389 

0.91 611 

9.99683 

5 

56 

9.08 176 

9.08  495 

106 

0.91 505 

9.99  681 

4 

57 

9.08  280 

104 

9.08  600 

105 

0.91  400 

9.99680 

3 

58 

9.08  383 

103 

9.08  705 

105 

0.91 295 

9.99  678 

2 

59 

9.08486 

103 
103 

9.08  810 

105 
104 

0.91 190 

9.99677 

1 

60 

9.08  589 

9.08  914 

0.91  086 

9.99  676 

0 

LGos 

d 

LCtn 

Cd 

LTan 

L  Sin 

/ 

Prop.  Pts.             1 

83°  — Logarithms  of  Trigonometric  Functions 


Ill] 


T  —  Logarithms  of  Trigonometric  Functions 


53 


L  Sin 


LTan 


cd       LCtn 


LGos 


Prop.  Pts. 


0 

9.08  589 

1 

9.08  692 

2 

9.08  795 

3 

9.08  897 

4 

9.08  999 

5 

9.09 101 

() 

9.09  202 

7 

9.09  304 

8 

9.09  405 

9 

9.09506 

10 

9.09  606 

11 

9.09  707 

12 

9.09  807 

13 

9.09  907 

14 

9.10  006 

15 

9.10106 

16 

9.10  205 

17 

9.10  304 

18 

9.10402 

19 

9.10  501 

20 

9.10  599 

21 

9.10  697 

oo 

9.10  795 

23 

9.10  893 

24 

9.10  990 

25 

9.11  087 

26 

9.11 184 

27 

9.11  281 

28 

9.11  377 

29 

9.11474 

30 

9,11 570 

31 

9.11 666 

32 

9.11  761 

33 

9.11  857 

34 

9.11  952 

35 

9.12  047 

36 

9.12  142 

37 

9.12  236 

38 

9.12  331 

39 

9.12  425 

40 

9.12  519 

41 

9.12  612 

42 

9.12  706 

43 

9.12  799 

44 

9.12  892 

45 

9.12  985 

46 

9.13  078 

47 

9.13171 

48 

9.13  263 

49 

9.13  355 

50 

9.13447 

51 

9.13  539 

62 

9.13  630 

53 

9.13  722 

54 

9.13  813 

55 

9.13  904 

56 

9.13  994 

57 

9.14085 

58 

9.14 175 

59 

9.14  266 

60 

9.14  356 

103 
103 
102 
102 
102 
101 
102 
101 
101 
100 
101 
100 
100 
99 
100 
99 
99 


9.08  914 

9.09  019 
9.09  123 
9.09  227 
9.09  330 
9.09  434 
9.09  537 
9.09  640 
9.09  742 
9.09  845 

9.09  947 
9.10049 

9.10  150 
9.10  252 
9.10  353 
9.10  454 
9.10  555 
9.10  656 
9.10  756 
9.10  856 
9.10  956 

11056 

11  155 

11254 

11353 

11452 

11551 

11649 

11747 

11845 

Ml  943 

1.12  040 

f.12138 

1.12  235 

1.12  332 

M2  428 

1.12  525 

1.12  621 
M2  717 
>.12  813 

).12  909 

1.13  004 
1.13099 
M3  194 
1.13  289 


9.13  384 
9.13  478 
9.13  573 
9.13  667 
9.13  761 
9.13  854 

9.13  948 

9.14  041 
9.14134 
9.14  227 
9.14  320 
9.14  412 
9.14  504 
9.14  597 
9.14  688 
9.14  780 


0.91 086 
0.90  981 
0.90  877 
0.90  773 
0.90  670 
0.90  566 
0.90  463 
0.90  360 
0.90  258 
0.90  155 
0.90  053 
0.89  951 
0.89  850 
0.89  748 
0.89647 
0.89  546 
0.89  445 
0.89  344 
0.89  244 
0.89  144 
0.89  044 
0.88  944 
0.88  845 
0.88  746 
0.88647 
0.88  548 
0.88  449 
0.88  351 
0.88  253 
0.88  155 
0.88057 
0.87  960 
0.87  862 
0.87  765 
0.87  668 
0.87  572 
0.87  475 
0.87  379 
0.87  283 
0.87  187 
0.87  091 
0.86  996 
0.86  901 
0.86  806 
0.86  711 
0.86  616 
0.86  522 
0.86  427 
0.86  333 
0.86  239 
0.86  146 
0.86052 
0  85  959 
0.85  866 
0.85  773 
0.85  680 
0.85  588 
0.85  496 
0.85  403 
0.85  312 
0.85  220 


9.99  675 
9.99  674 
9.99  672 
9.99670 
9.99  669 
9.99  667 
9.99  666 
9.99664 
9.99  663 
9.99  661 
9.99659 
9.99  658 
9.99  656 
9.99  655 
9.99  653 
9.99651 
9.99  650 
9.99648 
9.99  647 
9.99  645 
9.99  643 
9.99  642 
9.99  640 
9.99  638 
9.99  637 
9.99  635 
9.99  633 
9.99  632 
9.99630 
9.99  629 
9.99627 
9.99625 
9.99624 
9.99622 
9.99  620 
9.99618 
9.99  617 
9.99  615 
9.99  613 
9.99  612 
9.99  610 
9.99  608 
9.99  607 
9.99  605 
9.99  603 
9.99  601 
9.99  600 
9.99  598 
9.99  596 
9.99  595 
9.99  593 
9.99  591 
9.99  589 
9.99  588 
9.99  586 
9.99  584 
9.99  582 
9.99581 
9.99  579 
9.99  577 
9.99  575 


2 

105 

21.0 

104 

20.8 

103 

20.6 

3 

31.5 

31.2 

30.9 

4 

42.0 

41.6 

41.2 

5 

52.5 

52.0 

51.5 

6 

63.0 

62.4 

61.8 

7 

73.5 

72.8 

72.1 

8 

84.0 

83.2 

82.4 

9 

94.5 

93.6 

92.7 

101 

99 

98 

20.2 

19.8 

19.6 

30.3 

29.7 

29.4 

40.4 

39.6 

39.2 

50.5 

49.5 

49.0 

60.6 

59.4 

58.8 

70.7 

69.3 

68.6 

80.8 

79.2 

78.4 

90.9 

89.1 

88.2 

96 

95 

94 

19.2 

19.0 

18.8 

28.8 

28.5 

28.2 

38.4 

38.0 

37.6 

48.0 

47.5 

47.0 

57.6 

57.0 

56.4 

67.2 

66.5 

65.8 

76.8 

76.0 

75.2 

86.4 

85.5 

84.6 

20.4 
30.6 
40.8 
51.0 
61.2 
71.4 
81.6 
91.8 


97 

19.4 
29.1 
38.8 
48.5 
58.2 
67.9 
77.6 
87.3 


93 

18.6 
27.9 
37.2 
46.5 
55.8 
65.1 
74.4 
83.7 


92 

91 

2 

18.4 

18.2 

3 

27.6 

27.3 

4 

36.8 

36.4 

5 

46.0 

45.5 

6 

55.2 

54.6 

7 

64.4 

63.7 

8 

73.6 

72.8 

9 

82.8 

81.9 

90 

18.0 
27.0 
36.0 
45.0 
54.0 
63.0 
72.0 
81.0 


From  the  top : 

For  7°+  or  187°+, 
read  as  printed  ;  for 
97°+  or  277^^-,  read 
co-function. 

From  the  bottom  : 

For  82°+  or  262°+, 

read  as  printed  ;  for 
172°+  or  352°+,  read 
CQ-function. 


LGos 


LCtn 


c  d       L  Tan 


L  Sin 


Prop.  Pts. 


82°— Logarithms  of  Trigonometric  Functions 


54 


8°  — Logarithms  of  Trigonometric  Functions 


[in 


L  Sin 


L  Tan     c  d      L  Ctn 


L  Cos 


Prop.  Pts. 


10 

11 

12 
13 
14 
15 

16 

17 
18 
19 

20 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


9.14  356 
9.14  445 
9.14  535 
9.14  624 
9.14  714 
9.14  803 
9.14  891 

9.14  980 
9cl5  069 

9.15  157 
9.15  245 
9.15  333 
9.15  421 
9.15  508 
9.15  596 
9.15  683 
9.15  770 
9.15  857 

9.15  944 

9.16  030 
9.16  116 
9.16  203 
9.16  289 
9.16  374 
9.16  460 
9.16  545 
9.16  631 
9.16  716 
9.16  801 
9.16  886 

9.16  970 

9.17  055 
9.17  139 
9.17  223 
9.17  307 
9.17  391 
9.17  474 
9.17  558 
9.17  641 
9.17  724 
9.17  807 
9.17  890 

9.17  973 

9.18  055 
9.18137 
9.18  220 
9.18  302 
9.18  383 
9.18  465 
9.18  547 
9.18  628 
9.18  709 
9.18  790 
9.18  871 

9.18  952 
9.19033 
9.19 113 

9.19  193 
9.19  273 
9.19  353 
9.19433 


9.14  780 
9.14  872 

9.14  963 

9.15  054 
9.15  145 
9.15  236 
9.15  327 
9.15  417 
9.15  508 
9.15  598 
9.15  688 
9.15  777 
9.15  867 

9.15  956 

9.16  046 
9.16  135 
9.16  224 
9.16  312 
9.16  401 
9.16  489 
9.16  577 
9.16  665 
9.16  753 
9.16  841 

9.16  928 

9.17  016 
9.17  103 
9.17  190 
9.17  277 
9.17  363 
9.17  450 
9.17  536 
9.17  622 
9.17  708 
9.17  794 
9.17  880 

9.17  965 

9.18  051 
9-18  136 
9.18  221 
9.18  306 
9.18  391 
9.18  475 
a  18  560 
9.18  644 
9.18  728 
9.18  812 
9.18  896 

9.18  979 
9.19063 
9.19 146 

9.19  229 
9,19  312 
9.19  395 
9.19  478 
9.19  561 
9.19  643 
9.19  725 
9.19  807 
9.19  889 
9.19  971 


0.85  220 
0.85  128 
0.85  037 
0.84  946 
0.84  855 
0.84  764 
0.84  673 
0.84  583 
0.84  492 
0.84  402 
0.84  312 
0.84  223 
0.81 133 
0.84  044 
0.83  954 
0.83  865 
0.83  776 
0.83  688 
0.83  599 
0.83  511 
0.83  423 
0.83  335 
0.83  247 
0.83  159 
0.83  072 
0.82  984 
0.82  897 
0.82  810 
0.82  723 
0.82  637 
0,82  550 
0.82  464 
0.82  378 
0.82  292 
0.82  206 
0.82  120 
0.82  035 
0.81  949 
0.81  864 
0.81  779 
0.81 694 
081609 
0.81  525 
0.81  440 
0.81  356 
0.81 272 
0-81  188 
0.81 104 
0.81  021 
0.80  937 
0.80  854 
0.80  771 
0.80  688 
0.80  605 
0.80  522 
0.80  439 
0.80  357 
0.80  275 
0.80  193 
0.80  111 
0.80  029 


9.99  575 
9.99  574 
9.99  572 
9.99  570 
9.99  568 
9.99  566 
9.99  565 
9.99  563 
9.99  561 
9.99  559 
9.99  557 
9.99  556 
9.99  554 
9.99  552 
9.99  550 
9.99  548 
9.99  546 
9.99  545 
9.99  543 
9.99  541 
9.99  539 
9.99  537 
9.99  535 
9.99  533 
9.99  532 
9.99  530 
9.99  528 
9.99  526 
9.99  524 
9.99  522 
9.99  520 
9.99  518 
9.99517 
9.99515 
9.99  513 
9.99  511 
9.99  509 
9.99  507 
9.99  505 
9.99  503 
9.99  501 
9.99  499 
9.99497 
9.99  495 
9.99494 
9.99  492 
9.99490 
9.99  488 
9.99  486 
9.99484 
9.99  482 
9.99480 
9.99  478 
9.99  476 
9.99  474 
9.99  472 
9.99  470 
9.99  468 
9.99  466 
9.99  464 
9.99462 


92 

91 

90 

18.4 

18.2 

18.0 

27.6 

27.3 

27.0 

3(18 

36.4 

36.0 

460 

45.5 

45.0 

55.2 

54.6 

54.0 

64.4 

63.7 

63.0 

73.6 

72.8 

72.0 

82.8 

81.9 

81.0 

89 

17.8 
26.7 
35.6 
44.5 
53.4 
62.3 
71.2 
80.1 


88 

87 

17.6 

17.4 

26.4 

26.1 

35.2 

34.8 

44.0 

43.5 

52.8 

52.2 

61.6 

60.9 

70.4 

69.6 

79.2 

78.3 

85 

84 

17.0 

16.8 

25.5 

25.2 

34.0 

33.6 

42.5 

42.0 

51.0 

50.4 

59.5 

58.8 

68.0 

67.2 

76.5 

75.6 

86 

17.2 

25.8 
34.4 
43.0 
51.6 
60.2 
68.8 
77.4 

83 

16.6 
24.9 
33.2 
41.5 

49.8 
58.1 
66.4 

74.7 

80 

16.0 
24.0 
32.0 
400 
48.0 
56.0 
64.0 
72.0 


From  the  top : 

For  8°+ or  188°+,  read 
as  printed  ;  for  98°+  or 
278°+,  read  co-function. 

From  the  bottom : 

For  81°+  or  261°+, 
read  as  printed  ;  for 
171°+  or  351°+,  read 
co-function. 


82 

81 

2 

16.4 

16.2 

3 

24.6 

24.3 

4 

32.8 

32.4 

5 

41.0 

40.5 

6 

49.2 

48.6 

7 

57.4 

66.7 

8 

65.6 

64.8 

9 

73.8 

72.9 

LGos 


L  Ctn      c  d 


L  Tan 


L  Sin 


Prop.  Pts. 


8F  —  Logarithms  of  Trigonometric  Functions 


IIIJ 


9°  —  Logarithms  of  Trigonometric  Functions 


55 


LSin 


L  Tan    led     L  Ctn 


LCos 


Prop.  Pts. 


0 

1 

2 

3 

4 

6 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

k; 

17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
3() 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


9.19  433 
9.19  513 
9.19  592 
9.19  672 
9.19  751 
9.19  830 
9.19  909 

9.19  988 

9.20  067 
9.20  145 
9.20  223 
9.20  302 
9.20  380 
9.20  458 
9.20  535 
9.20  613 
9.20  691 
9.20  7(;8 
9.20  845 
9.20  922 

9.20  9m 

9.21  076 
9.21 153 
9.21  229 
9.21  306 
9.21  382 
9.21 458 
9.21  534 
9.21  610 
9.21  685 
9.21  761 
9.21  836 
9.21  912 

9.21  987 

9.22  062 
9.22  137 
9.22  211 
9.22  286 
9.22  361 
9.22  435 
9.22  509 
9.22  583 
9.22  657 
9.22  731 
9.22  805 
9.22  878 

9.22  952 

9.23  025 
9.23  098 
9.23  171 
9.23  244 
9.23 .317 
9.23  390 
9.23  462 
9.23  535 
9.23  607 
9.23679 
9.23  752 
9.23  823 
9.23  895 
9.23  967 


9.19971 
9.20  0.53 
9.20  lU 
9.20  216 
9.20  297 
9.20  378 
9.20  459 
9.20  540 
9.20  621 
9.20  701 
9.20  782 
9.20  862 

9.20  942 
9.21022 
9.21 102 
9.21 182 

9.21  261 
9.21  341 
9.21  420 
9.21  499 
9.21  578 
9.21  657 
9.21  7;3(> 
9.21814 
9.21893 

9.21  971 

9.22  049 
9.22  127 
9.22  205 
9.22  283 
9.22  361 
9.22  438 
9.22  516 
9.22  593 
9.22  670 
9.22  747 
9.22  824 
9.22  901 

9.22  977 

9.23  054 
9.23130 
9.23  206 
9.23  283 
9.23  359 
9.23435 
9.23  510 
9.23  586 
9.23  661 
9.23  737 
9.23  812 
9.23  887 

9.23  962 

9.24  037 
9.24  112 
9.24  186 
9.24  261 
9.24  335 
9.24  410 
9.24  484 
9.24  558 
9.24  632 


0.80  029 
0.79  <H7 
0.79  866 
0.79  784 
0.79  703 
0.79  622 
0.79  541 
0.79460 
0.79  379 
0.79  299 
0.79  218 
0.79  138 
0.79  058 
0.78  978 
0.78  898 
0.78  818 
0.78  739 
0.78  659 
0.78  580 
0.78  501 
0.78  422 
0.78  343 
0.78  264 
0.78  186 
0.78  107 
0.78  029 
0.77  951 
0.77  873 
0.77  795 
0.77  717 
0.77  639 
0.77  562 
0.77  484 
0.77  407 
0.77  330 
0.77  253 
0.77  176 
0.77  099 
0.77  023 
0.76  946 
0.76  870 
0.76  71^)4 
0.76  717 
0.76  641 
0.76  565 
0.76  490 
0.76  414 
0.76  339 
0.76  263 
0.76 188 
0.76113 
0.76  038 
0.75  963 
0.75  888 
0.75  814 
0.75  739 
0.75  665 
0.75  590 
0.75  516 
0.75  442 
0.75  368 


9.99  462 
9.99  460 
9.99  458 
9.99456 
9.99  454 
9.99  452 
9.99  450 
9.99  448 
9.99  446 
9.99444 
9.99442 
9.99  440 
9.99438 
9.99  436 
9.99434 
9.99  432 
9.99  429 
9.99427 
9.99  425 
9.99  423 
9.99421 
9.99  419 
9.9.0  417 
9.99  415 
9.99  413 
9.99  411 
9.99409 
9.99  407 
9.<)9  404 
9.99  402 
9.99  400 
9.99  398 
9.99  396 
9.99  394 
9.99  392 
9.99  390 
9.99  388 
9.99  385 
9.99  383 
9.99381 
9.99  379 
9.99  377 
9.99  375 
9.99  372 
9.99  370 
9.99  368 
9.99  366 
9.99  364 
9.99  362 
9.99  359 
9.99  357 
9.99  355 
9.99  353 
9.99  351 
9.99  348 
9.99  346 
9.99  344 
9.99  342 
9.99  340 
9.99  337 
9.99  335 


82 

81 

80 

16.4 

16.2 

16.0 

24.6 

24.3 

24.0 

32.8 

32.4 

32.0 

41.0 

40.5 

40.0 

49.2 

48.6 

48.0 

57.4 

56.7 

56.0 

65.6 

64.8 

64.0 

73.8 

72.9 

72.0 

78 

77 

76 

2 

15.6 

15.4 

15.2 

3 

23.4 

23.1 

22.8 

4 

31.2 

30.8 

30.4 

5 

39.0 

38.5 

38.0 

6 

46.8 

46.2 

45.6 

7 

54.6 

53.9 

53.2 

8 

62.4 

61.6 

60.8 

9 

70.2 

69.3 

68.4 

74 

73 

72 

14.8 

14.6 

14.4 

22.2 

21.9 

21.6 

29.6 

29.2 

28.8 

37.0 

36.5 

36.0 

44.4 

43.8 

43.2 

51.8 

51.1 

50.4 

59.2 

58.4 

57.6 

66.6 

65.7 

64.8 

79 

15.8 
23.7 
31.6 
39.5 
47.4 
55.3 
63.2 
71.1 


75 

15.0 
22.5 
30.G 
37.5 
45.0 
52.5 
60.0 
67.5 


71 

14.2 
21.3 
28.4 
35.5 
42.6 
49.7 
56.8 
63.9 


From  the  top: 

For  9^+,  or  189'^+,  read 
as  printed  ;  for  99°+  or 
279°+,  read  co-functiou. 

From  the  bottom: 

For    80°+    or    260°+, 

read  as  printed  ;  for 
170°+  or  350°+,  read 
co-function. 


L  Cos 


LCtn 


cd 


LTan 


L  Sin 


Prop.  Pts. 


80"^— Logarithms  of  Trigonometric  Functions 


56  10°— Logarithms  of  Trigonometric  Functions         [iii 


L  Sin 


L  Tan     c  d      L  Gtn 


LCos 


Prop.  Pts. 


0 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


9.23  967 

9.24  039 
9.24  110 
9.24  181 
9.24  253 
9.24  324 
9.24  395 
9.24  466 
9.24  536 
9.24607 
9.24  677 
9.24  748 
9.24  818 
9.24  888 

9.24  958 

9.25  028 
9.25  098 
9.25  168 
9.25  237 
9.25  307 
9.25  376 
9.25  445 
9.25  514 
9.25  583 
9.25652 
9.25  721 
9.25  790 
9.25  858 
9.25  927 

9.25  995 

9.26  063 
9.26  131 
9.26  199 
9.26  267 
9.26  335 
9.26  403 
9.26  470 
9.26  538 
9.26  605 
9.26  672 
9.26  739 
9.26  806 
9.26  873 

9.26  940 

9.27  007 
9.27  073 
9.27  140 
9.27  206 
9.27  273 
9.27  339 
9.27  405 
9.27  471 
9.27  537 
9.27  602 
9.27  668 
9.27  734 
9.27  799 
9.27  864 
9.27  930 

9.27  995 

9.28  060 


9.24  632 
9.24  706 
9.24  779 
9.24  853 

9.24  926 

9.25  000 
9.25  073 
9.25  146 
9.25  219 
9.25  292 
9.25  365 
9.25  437 
9.25  510 
9.25  582 
9.25  655 
9.25  727 
9.25  799 
9.25  871 

9.25  943 

9.26  015 
9.26  086 
9.26  158 
9.26  229 
9.26  301 
6.26  372 
9.26  443 
9.26  514 
9.26  585 
9.26  655 
9.26  726 
9.26  797 
9.26  867 

9.26  937 

9.27  008 
9.27  078 
9.27  148 
9.27  218 
9.27  288 
9.27  357 
9.27  427 
9.27  496 
9.27  566 
9.27  635 
9.27  704 
9.27  773 
9.27  842 
9.27  911 

9.27  980 

9.28  049 
9.28  117 
9.28  186 
9.28  254 
9.28  323 
9.28  391 
9.28  459 
9.28  527 
9.28  595 
9.28  662 
9.28  730 
9.28  798 
9.28  865 


0.75  368 
0.75  294 
0.75  221 
0.75 147 
0.75  074 
0.75  000 
0.74  927 
0.74  854 
0.74  781 
0.74  708 
0.74  635 
0.74  563 
0.74  490 
0.74  418 
0.74  345 
0.74  273 
0.74  201 
0.74  129 
0.74  057 
0.73  985 
0.73  914 
0.73  842 
0.73  771 
0.73  699 
0.73  628 
0.73  557 
0.73  486 
0.73  415 
0.73  345 
0.73  274 
0.73  203 
0.73 133 
0.73  063 
0.72  992 
0.72  922 
0.72  852 
0.72  782 
0.72712 
0.72  643 
0.72  573 
0.72  504 
0.72  434 
0.72  365 
0.72  296 
0.72  227 
0.72  158 
0.72  089 
0.72  020 
0.71  951 
0.71  883 
0.71  814 
0.71  746 
0.71 677 
0.71 609 
0.71  541 
0.71  473 
0.71 405 
0.71 338 
0.71  270 
0.71  202 
0.71 135 


9.99  335 
9.99  333 
9.99  331 
9.99  328 
9.99  326 
9.99  324 
9.99  322 
9.99  319 
9.99  317 
9.99  315 
9.99  313 
9.99  310 
9.99  308 
9.99  306 
9.99  304 
9.99  301 
9.99  299 
9.99  297 
9.99  294 
9.99  292 
9.99  290 
9.99  288 
9.99  285 
9.99  283 
9.99  281 
9.99  278 
9.99  276 
9.99  274 
9.99  271 
9.99269 

9.99  267 
9.99  264 
9.99  262 
9.99  260 
9.99257 
9.99  255 
9.99  252 
9.99  250 
9.99  248 
9.99  245 
9.99  243 
9.99  241 
9.99  238 
9.99  236 
9.99233 
•9.99  231 
9.99  229 
9.99  226 
9.99  224 
9.99  221 
9.99  219 
9.99  217 
9.99  214 
9.99  212 
9.99  209 
9.99  207 
9.99  204 
9.99  202 
9.99  200 
9.99  197 
9.99  195 


74 

73 

2 

14.8 

14.6 

3 

22.2 

21.9 

4 

29.6 

29.2 

5 

37.0 

36.5 

6 

44.4 

43.8 

7 

51.8 

51.1 

8 

59.2 

58.4 

9 

66.6 

65.7 

71 

70 

2 

14.2 

14.0 

3 

21.3 

21.0 

4 

28.4 

28.0 

5 

35.5 

35.0 

6 

42.6 

42.0 

7 

49.7 

49.0 

8 

56.8 

56.0 

9 

63.9 

63.0 

68 

67 

2 

13.6 

13.4 

3 

20.4 

20.1 

4 

27.2 

26.8 

5 

34.0 

33.5 

6 

40.8 

40.2 

7 

47.6 

46.9 

8 

54.4 

53.6 

9 

61.2 

60.3 

72 

14.4 
21.6 
28.8 
36.0 
43.2 
50.4 
57.6 
64.8 


69 

13.8 
20.7 
27.6 
34.5 
41.4 
48.3 
55.2 
62.1 

66 

13.2 
19.8 
26.4 
33.0 
39.6 
46.2 
52.8 
59.4 


65 

2 

13.0 

3 

19.5 

4 

26.0 

5 

32.5 

6 

39.0 

7 

45.5 

8 

52.0 

9 

58.5 

0.6 
0.9 
1.2 
1.5 
1.8 
2.1 
2.4 
2.7 


From  the  top: 

For  10°+  or  190°+, 

read  as  printed  ;  for 
100°+  or  280°+,  read 
co-function. 

From  the  bottom : 

For  79°+  or  269°+, 
read  as  printed;  for 
169°+  or  349°+,  read 
co-function. 


LCos 


L  Ctn      c  d 


L  Tan 


L  Sin      d      ' 


Prop.  Pts. 


79°  —  Logarithms  of  Trigonometric  Functions 


11°  —  Logarithms  of  Trigonometric  Functions 


57 


LSin 


L  Tan     c  d      L  Gtn 


L  Cos 


Prop.  Pts. 


0 

1 
2 

3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
'57 
58 
59 
60 


9.28  060 
9.28  125 
9.28  190 
9.28  254 
9.28  319 
9.28  384 
9.28  448 
9.28  512 
9.28  577 
9.28  641 
9.28  705 
9.28  769 
9.28  833 
9.28  896 

9.28  960 

9.29  024 
9.29  087 
9.29  150 
9.29  214 
9.29  277 
9.29  340 
9.29  403 
9.29  466 
9.29  529 
9.29  591 
9.29654 
9.29  716 
9.29  779 
9.29  841 
9.29  903 

9.29  966 

9.30  028 
9.30  090 
9.30  151 
9.30  213 
9.30  275 
9.30  336 
9.30  398 
9.30  459 
9.30  521 
9.30  582 
9.30  643 
9.30  704 
9.30  765 
9.30  826 
9.30  887 

9.30  947 
9.31 008 
9.31068 
9.31 129 
9.31 189 

9.31  250 
9.31  310 
9.31  370 
9.31 430 
9.31  490 
9.31 549 
9.31 609 
9.31  669 
9.31  728 
9.31  788 


9.28  865 

9.28  933 

9.29  000 
9.29  067 
9.29 134 
9.29  201 
9.29  268 
9.29  335 
9.29  402 
9.29  468 
9.29  535 
9.29601 
9.29668 
9.29  734 
9.29  800 
9.29  866 
9.29  932 

9.29  998 

9.30  064 
9.30  130 
9.30  195 
9.30  261 
9.30  326 
9.30  391 
9.30  457 
9.30  522 
9.30  587 
9.30  652 
9.30  717 
9.30  782 
9.30  846 
9.;30  911 

9.30  975 

9.31  040 
9.31 104 
9.31 168 
9.31  233 
9.31  297 
9.31  361 
9.31  425 
9.31  489 
9.31  552 
9.31  616 
9.31  679 
9.31  743 
9.31  806 
9.31  870 

9.31  933 
9.31 996 

9.32  059 
9.32  122 
9.32  185 
9.32  248 
9.32  311 
9.32  373 
9.32  436 
9.32  498 
9.32  561 
9.32  623 
9.32  685 
9.32  747 


71 135 

71  067 

71000 

70  933 

70  866 

70  799 

70  732 

,70  665 

,70  598 

70  532 

,70465 

,70  399 

70  332 

,70  266 

,70  200 

0.70  134 

0.70068 

0.70  002 

0.69  936 

0.69  870 

0.69  805 

0.69  739 

0.69  674 

0.69  609 

0.69  543 

0.69  478 

0.69  413 

0.69  348 

0.69  283 

0.69  218 

0.69 154 

0.69089 

0.69  025 

0.68  960 

0.68  896 

0.68  832 

0.68  767 

0.68  703 

0.68  639 

0.68  575 

0.68  511 

0.68  448 

0.68  384 

0.68  321 

0.68  257 

0.68 194 

0.68  130 

0.68  067 

0.68  004 

0.67  941 

0.67  878 

0.67  815 

0.67  752 

0.67  689 

0.67  627 

0.67  564 

0.67  502 

0.67  439 

0.67  377 

0.67  315 

0.67  253 


9.99 195 
9.99 192 
9.99 190 
9.99  187 
9.99  185 
9.99 182 
9.99  180 
9.99177 
9.99  175 
9.99 172 
9.99170 
9.99  167 
9.99 165 
9.99 162 
9.99  160 
9.99 157 
9.99  155 
9.99 152 
9.99 150 
9.99  147 
9.99  145 
9.99  142 
9.99140 
9.99  137 
9.99  135 
9.99 132 
9.99  130 
9.99  127 
9.99  124 
9.99 122 
9.99 119 
9.99  117 
9.99 114 
9,99 112 
9.99  109 
9.99 106 
9.99  104 
9.99  101 
9.99  099 
9.99  096 
9.99093 
9.99  091 
9.99  088 
9.99  086 
9.99083 
9.99  080 
9.99  078 
9.99  075 
9.99072 
9.99  070 
9.99  067 
9.99  064 
9.99062 
9.99059 
9.99056 
9.99  054 
9.99  051 
9.99048 
9.99046 
9.99  043 
9.99  040 


68 

67 

2 

13.6 

13.4 

3 

20.4 

20.1 

4 

27.2 

26.8 

5 

34.0 

33.5 

6 

40.8 

40.2 

7 

47.6 

46.9 

8 

54.4 

53.6 

9 

61.2 

60.3 

65 

64 

2 

13.0 

12.8 

3 

19.5 

19.2 

4 

26.0 

25.6 

5 

32.5 

32.0 

6 

39.0 

38.4 

7 

45.5 

44.8 

8 

52.0 

51.2 

9 

58.5 

57.6 

62 

61 

2 

12.4 

12.2 

3 

18.6 

18.3 

4 

24.8 

24.4 

5 

31.0 

30.5 

6 

37.2 

36.6 

7 

43.4 

42.7 

8 

49.6 

48.8 

9 

55.8 

54.9 

66 

13.2 
19.8 
26.4 
33.0 
39.6 
46.2 
52.8 
59.4 

63 

12.6 
18.9 
25.2 
31.5 
37.8 
44.1 
50.4 
56.7 

60 

12.0 
18.0 
24.0 
30.0 
36.0 
42.0 
48.0 
54.0 


59 

2 

11.8 

3 

17.7 

4 

23.6 

5 

29.5 

6 

35.4 

7 

41.3 

8 

47.2 

9 

53.1 

3 

0.6 
0.9 
1.2 
1.5 
1.8 
2.1 
2.4 
2.7 


From  the  top  : 

For  11°+  or  191°+, 

read  as  printed  ;  for 
101°+  or  281°+, read 
co-function. 

Fi'om  the  bottom : 

For  78°+  or  258°+, 
read  as  printed  ;  for 
168°+  or  348°+,  read 
co-function. 


L  Cos 


LCtn 


c  d      L  Tan 


L  Sin    I  d      ' 


Prop.  Pts. 


78°  — Logarithms  of  Trigonometric  Functions 


5S 


12°  —  Logarithms  of  Trigonometric  Functions 


[111 


LSin 


L  Tan     c  d      L  Ctn 


L  Cos 


Prop.  Pts. 


10 

11 
12 


15 

16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 

35 

36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


9.31  788 
9.31  847 
9.31  907 

9.31  966 

9.32  025 
9.32  084 
9.32  143 
9.32  202 
9.32  261 
9.32  319 


9.32  378 
9.32  437 
9.32  495 

13  9.32  553 

14  9.32  612 


9.32  670 
9.32  728 
9.32  786 
9.32  844 
9.32  902 

9.32  960 

9.33  018 
9.33  075 
9.33  133 
9.33  190 
9.33  248 
9.33  305 
9.33  362 
9.33  420 
9.33  477 
9.33  534 
9.33  591 
9.33  647 
9.33  704 
9.33  761 
9.33  818 
9.33  874 
9.33  931 

9.33  987 

9.34  043 
9.34  100 
9.34  156 
9.34  212 
9.34  268 
9.34  324 
9.34  380 
9.34  4^^ 
9.34  491 
9.34  547 
9.34  602 


34658 
.34  713 
34  769 
.34  824 
34  879 
34  934 

34  989 

35  044 
35  099 
35154 
35  209 


9.32  747 
9.32  810 
9.32  872 
9.32  933 

9.32  995 

9.33  057 
9.33119 
9.33  180 
9.33  242 
9.33  303 
9.33  365 
9.33  426 
9.33  487 
9.33  548 
9.33  609 
9.33  670 
9.33  731 
9.33  792 
9.33  853 
9.33  913 

9.33  974 

9.34  034 
9.34  095 
9.34  155 
9.34  215 
9.34  276 
9.34  336 
9.34  396 
9.34  456 
9.34  516 
9.34  576 
9.34  635 
9.34  695 
9.34  755 
9.34  814 
9.34  874 
9.34  933 

9.34  992 

9.35  051 
9.35  111 
9.35  170 
9.35  229 
9.35  288 
9.35  347 
9.35  405 
9.35  464 
9.35  523 
9.35  581 
9.35  640 
9.35  698 

9.35  757 
9.35  815 
9.35  873 
9.35  931 

9.35  989 

9.36  047 
9.36  105 
9.36  163 
9.36  221 
9.36  279 
9.36  336 


0.67  253 
0.67  190 
0.67  128 
0.67  067 
0.67  005 
0.66  943 
0.66  881 
0.66  820 
0.66  758 
0.66  697 
0.66  635 
0.66  574 
0.66  513 
0.(36  452 
0.66  391 
0.66  330 
0.66  269 
0.66  208 
0.66  147 
0.66  087 
0.66  026 
0.65  966 
0.65  905 
0.65  845 
0.65  785 
0.65  724 
0.65  664 
0.65  604 
0.65  544 
0.65  484 
0.65  424 
0.(J5  365 
0.65  305 
0.65  245 
0.65  186 
0.65  126 
0.65  067 
0.65  008 
0.64  949 
0.64  889 
0.64  830 
0.64  771 
0.64  712 
0.64  653 
0.64  595 
0.64  536 
0.64  477 
0.64  419 
0.64  3()0 
0.64  302 
0.64  243 
0.64 185 
0.64127 
0.64  069 
0.64  011 
0.63  953 
0.63  895 
0.63  837 
0.63  779 
0.63  721 
0.63  664 


9.99  040 
9.99038 
9.99  035 
9.99  032 
9.99030 
9.99  027 
9.99  024 
9.99  022 
9.99  019 
9.99  016 
9.99  013 
9.99011 
9.99008 
9.99  005 
9.99  002 
9.99  000 
9.98  997 
9.98  994 
9.98  991 
9.98  989 
9.98  986 
9.98  983 
9.98  980 
9.98  978 
9.98  975 
9.98  972 
9.98  969 
9.98  967 
9.98  964 
9.98  961 
9.98  958 
9.98  955 
9.98  953 
9.98  950 
9.98  947 
9.98  944 
9.98  941 
9.98  938 
9.98  936 
9.98  933 
9.98  930 
9.98  927 
9.98  924 
9.98  921 
9.98  919 
9.98  916 
9.98  913 
9.98  910 
9.98  907 
9.98  904 
9.98  901 
9.98  898 
9.98  896 
9.98  893 
9.98  890 
9.98  887 
9.98  884 
9.98  881 
9.98  878 
9.98  875 
9.98  872 


63 

62 

2 

12.6 

12.4 

3 

18.9 

18.6 

4 

25.2 

24.8 

5 

31.5 

31.0 

6 

37.8 

37.2 

7 

44.1 

43.4 

8 

50.4 

49.6 

9 

56.7 

55.8 

60 

59 

2 

12.0 

11.8 

3 

18.0 

17.7 

4 

24.0 

23.6 

5 

30.0 

29.5 

6 

36.0 

35.4 

7 

42.0 

41.3 

8 

48.0 

47.2 

9 

54.0 

53.1 

61 

12.2 

18.3 
24.4 
30.5 
3().6 
42.7 
48.8 
54.9 


58 

11.6 
17.4 
23.2 
29.0 
34.8 
40.6 
46.4 
52.2 


57 

2 

11.4 

3 

17.1 

4 

22.8 

5 

28.5 

6 

34.2 

7 

39.9 

8 

45.6 

9 

51.3 

55 

2 

11.0 

3 

16.5 

4 

22.0 

5 

27.5 

6 

33.0 

7 

38.5 

8 

44.0 

9 

49.5 

56 

11.2 
16.8 
22.4 
28.0 
33.6 
39.2 
44.8 
50.4 


0.6 
0.9 
1.2 
1.5 
1.8 
2.1 
2.4 


From  the  top : 

For  12°+  or  192°+, 
read  as  printed;  for 
102°+  or  282°+,  read 
co-function. 

From  the  hottoryt : 

For  77°  or  257°, 
read  as  printed  ;  for 
167°  or  347°,  read 
co-function . 


LGos 


L  Ctn  I  c  d 


L  Tan 


L  Sin 


Prop.  Pts. 


77°  —  Logarithms  of  Trigonometric  Functions 


Ill]  13°  —  Logarithms  of  Trigonometric  Functions 


59 


'        LSin 


LTan 


c  d     L  Ctn 


L  Cos 


Prop.  Pts. 


9.35  209 
9.35  2(33 
9.35  318 
9.35  373 
9.35  427 
9.35  481 
9.35  536 
9.35  590 
9.35  ()44 
9.35  698 
9.35  752 
9.35  80(3 
9.35  860 
9.35  914 

9.35  968 

9.36  022 
9.36  075 
9.36  129 
9.36  182 
9.36  236 
9.36  289 
9.36  342 
9.36  395 
9.36  449 
9.36  502 
9.36  555 
9.36  (308 
9.36  660 
9.36  713 
9.36  766 
9.36  819 
9.36  871 
9.36  924 

9.36  976 

9.37  028 
9.37  081 
9.37  133 
9.37  185 
9.37  237 
9.37  289 
9.37  341 
9.37  393 
9.37  445 
9.37  497 
9.37  549 
9.37  600 
9.37  652 
9.37  703 
9.37  755 
9.37  806 
9.37  858 
9.37  909 

9.37  960 

9.38  011 
9.38  062 
9.38  113 
9.38  164 
9.38  215 
9.38  266 
9.38  317 
9.38  368 


9.36  336 
9.36  3f>4 
9.36  452 
9.36  509 
9.36  566 
9.36  624 
9.36  681 
9.36  738 
9.36  795 
9.36  852 
9.36  909 

9.36  966 

9.37  023 
9.37  080 
9.37  137 
9.37  193 
9.37  250 
9.37  306 
9.37  363 
9.37  419 
9.37  476 
9.37  532 
9.37  588 
9.37  644 
9.37  700 
9.37  756 
9.37  812 
9.37  868 
9.37  924 

9.37  980 

9.38  035 
9.38  091 
9.38  147 
9.38  202 
9.38  257 
9.38  313 
9.38  368 
9.38  423 
9.38  479 
9  38  534 
9.38  589 
9.38  (344 
9.38  699 
9.38  754 
9.38  808 
9.38  863 
9.38  918 

9.38  972 

9.39  027 
9.39  082 
9.39 136 
9.39  190 
9.39  245 
9.39  299 
9.39  353 
9.39407 
9.39  461 
9.39  515 
9.39  569 
9.39  623 
9.39  677 


0.63  6(34 
0.63  606 
0.63  548 
0.63  491 
0.63  434 
0.63  376 
0.63  319 
0.63  262 
0.63  205 
0.63  148 
0.63  091 
0.63  034 
0.62  977 
0.62  920 
0.62  863 
0.62  807 
0.62  750 
0.62  694 
0.62  637 
0.62  581 
0.62  524 
0.(32  468 
0.62  412 
0.62  356 
0.62  300 
0.62  244 
0.62  188 
0.62  132 
0.62  076 
0.62  020 
0.61  965 
0.61  909 
0.61  853 
0.61  798 
0.61  743 
0.61  687 
0.61  632 
0.61  577 
0.61  521 
0.61 466 
0.61  411 
0.61  3.56 
0.61  301 
0.61  246 
0.61 192 
0.61 137 
0.61 082 
0.61  028 
0.60  973 
0.60  918 
0.60  864 
0.60  810 
0.60  755 
0.60  701 
0.60  647 
0.60  593 
0.60  539 
0.60  485 
0.60  431 
0.60  377 
0.60  323 


9.98  872 
9.98  869 
9.98  867 
9.98  864 
9.98  861 
9.98  858 
9.98  855 
9.98  852 
9.98  849 
9.98  846 
9.98  843 
9.98  840 
9.98  837 
9.98  834 
9.98  831 
9.98  828 
9.98  825 
9.98  822 
9.98  819 
9.98  816 
9.98  813 
9.98  810 
9.98  807 
9.98  804 
9.98  801 
9.98  798 
9.98  795 
9.98  792 
9.98  789 
9.98  786 
9.98  783 
9.98  780 
9.98  777 
9.98  774 
9.98  771 
9.98  768 
9.98  765 
9.98  762 
9.98  759 
9.98  756 
9.98  753 
9.98  750 
9.98  746 
9.98  743 
9.98  740 
9.98  737 
9.98  734 
9.98  731 
9.98  728 
9.98  725 
9.98  722 
9.98  719 
9.98  715 
9.98  712 
9.98  709 
9.98  706 
9.98  703 
9.98  700 
9.98  697 
9.98694 
9.98  61X) 


58 

57 

2 

11.6 

11.4 

3 

17.4 

17.1 

4 

23.2 

22.8 

5 

29.0 

28.5 

6 

34.8 

34.2 

7 

40.6 

39.9 

8 

46.4 

45.6 

9 

52.2 

51.3 

55 

54 

2 

11.0 

10.8 

3 

16.5 

16.2 

4 

22.0 

21.6 

5 

27.5 

27.0 

6 

33.0 

32.4 

7 

38.5 

37.8 

8 

44.0 

43.2 

9 

49.5 

48.6 

56 

11.2 
16.8 
22.4 
28.0 
33.6 
39.2 
44.8 
50.4 


53 

10.6 
15.9 
21.2 
26.5 
31.8 
37.1 
42.4 
47.7 


52 

2 

10.4 

3 

15.6 

4 

20.8 

5 

26.0 

6 

31.2 

7 

36.4 

8 

41.6 

9 

46.8 

4 

2 

0.8 

3 

1.2 

4 

1.6 

5 

2.0 

6 

2.4 

7 

2.8 

8 

3.2 

9 

3.6 

51 

10.2 
15.3 
20.4 
25.5 
30.6 
35.7 
40.8 
45.9 


0.6 
0.9 
1.2 
1.5 
1.8 
2.1 
2.4 
2.7 


From  the  top : 

For  13°+ or  193^+, 

read  as  printed  ;  for 
103°+ or  283°+,  read 
co-function. 

From  the  bottom: 

For  76°  or:  256°, 

read  as  printed  ;  for 
166°+  or  346°+,  read 
co-function. 


LCDS 


LCtn 


c  d     L  Tan 


L  Sin 


d      f 


Prop.  Pts. 


76°— Logarithms  of  Trigonometric  Functions 


60 


14°  — Logarithms  of  Trigonometric  Functions         [in 


10 

11 

12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


L  Sin 


9.38  3G8 
9.38  418 
9.38  469 
9.38  519 
9.38  570 
9.38  620 
9.38  670 
9.38  721 
9.38  771 
9.38  821 
9.38  871 
9.38  921 

9.38  971 
9.39021 
9.39071 
9.39 121 
9.39 170 

9.39  220 
9.39270 
9.39319 
9.39  369 
9.39418 
9.39467 
9.39  517 
9.39  566 
9.39615 
9.39  664 
9.39  713 
9.39  762 
9.39  811 
9.39  860 
9.39  909 

9.39  958 

9.40  006 
9.40  055 
9.40 103 
9.40 152 
9.40  200 
9.40  249 
9.40  297 
9.40  346 
9.40  394 
9.40442 
9.40  490 
9.40  538 
9.40  586 
9.40  634 
9.40  682 
9.40  730 
9.40  778 
9.40  825 
9.40  873 
9.40  921 

9.40  968 
9.41 016 

9.41  063 
9.41  111 
9.41 158 
9.41  205 
9.41  252 
9.41  300 


L  Tan  c  d  L  Gtn 


9.39  677 
9.39  731 
9.39  785 
9.39  838 
9.39  892 
9.39  945 

9.39  999 

9.40  052 
9.40 106 
9.40  159 
9.40  212 
9.40  266 
9.40  319 
9.40  372 
9.40  425 
9.40  478 
9.40  531 
9.40  584 
9.40  636 
9.40  689 
9.40  742 
9.40  795 
9.40  847 
9.40  900 

9.40  952 
9.41 005 
9.41 057 
9.41 109 
9.41 161 

9.41  214 
9.41  266 
9.41  318 
9.41  370 
9.41  422 
9  41 474 
9.41  526 
9.41  578 
9.41  629 
9.41  681 
9.41 733 
9.41  784 
9.41  836 
9.41  887 
9.41  939 

9.41  990 

9.42  041 
9.42  093 
9.42  144 
9.42  195 
9.42  246 
9.42  297 
9.42  348 
9.42  399 
9.42  450 
9.42  501 
9.42  552 
9.42  603 
9.42  653 
9.42  704 
9.42  755 
9.42  805 


0.60  323 
0.60  269 
0.60  215 
0.60  162 
0.60  108 
0.60  055 
0.60  001 
0.59  948 
0.59  894 
0.59  841 
0.59  788 
0.59  734 
0.59  681 
0.59  628 
0.59  575 
0.59  522 
0.59  469 
0.59  416 
0.59  364 
0.59  311 
0.59  258 
0.59  205 
0.59  153 
0.59  100 
0.59048 
0.58  995 
0.58  943 
0.58  891 
0.58  839 
0.58  786 
0.58  734 
0.58  682 
0.58  630 
0.58  578 
0.58  526 
0.58  474 
0.58  422 
0.58  371 
0.58  319 
0.58  267 
0.58  216 
0.58  164 
0.58  113 
0.58  061 
0.58  010 
0.57  959 
0.57  907 
0.57  856 
0.57  805 
0.57  754 
0.57  703 
0.57  652 
0.57  601 
0.57  550 
0.57  499 
0.57  448 
0.57  397 
0.57  347 
0.57  296 
0.57  245 
0.57  195 


LGos 


9.98  690 
9.98  687 
9.98  684 
9.98  681 
9.98  678 
9.98  675 
9.98  671 
9.98  668 
9.98  665 
9.98  662 
9.98  659 
9.98  656 
9.98  652 
9.98  649 
9.98  646 
9.98  643 
9.98  640 
9.98  636 
9.98  633 
9.98  630 
9.98  627 
9.98  623 
9.98  620 
9.98  617 
9.98  614 
9.98  610 
9.98  607 
9.98  604 
9.98  601 
9.98  597 
9.98  594 
9.98  591 
9.98  588 
9.98  584 
9.98  581 
9.98  578 
9.98  574 
9.98  571 
9.98  568 
9.98  565 
9.98  561 
9.98  558 
9.98  555 
9.98  551 
9.98  548 
9.98  545 
9.98  541 
9.98  538 
9.98  535 
y.98  531 
9.98  528 
9.98  525 
9.98  521 
9.98  518 
9.98  515 
9.98  511 
9.98  508 
9.98  505 
9.98  501 
9.98  498 
9.98  494 


60 

59 
58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 

25 

24 
23 
22 
21 

20 

19 
18 
17 
16 
15 
14 
13 
12 
11 
10 
9 


Prop.  Pts. 


54 

53 

2 

10.8 

10.6 

3 

16.2 

15.9 

4 

21.6 

21.2 

5 

27.0 

26.5 

6 

32.4 

31.8 

7 

37.8 

37.1 

8 

43.2 

42.4 

9 

48.6 

47.7 

51 

50 

2 

10.2 

10.0 

3 

15.3 

15.0 

4 

20.4 

20.0 

6 

25.5 

25.0 

6 

30.6 

30.0 

7 

35.7 

35.0 

8 

40.8 

40.0 

9 

45.9 

45.0 

52 

10.4 
15.6 
20.8 
26.0 
31.2 
36.4 
41.6 
46.8 

49 

9.8 
14.7 
19.6 
24.5 
29.4 
34.3 
39.2 
44.1 


48 

2 

9.6 

3 

14.4 

4 

19.2 

6 

24.0 

6 

28.8 

7 

33.6 

8 

38.4 

9 

43.2 

4 

2 

0.8 

3 

1.2 

4 

1.6 

5 

2.0 

6 

2.4 

7 

2.8 

8 

3.2 

9 

3.6 

47 

9.4 
14.1 
18.8 
23.5 
28.2 
32.9 
37.6 
42.3 


0.6 
0.9 
1.2 
1.5 
1.8 
2.1 
2.4 
2.7 


From  the  top  : 

For  14°+  or  194°+, 
read  as  printed ;  for 
104°+  or  284°+,  read 

co-function.  ' 

From  the  bottom : 

For  75°+  or  255°+, 
read  as  printed;  for 
165°+  or  345°+,  read 
co-function. 


LGos 


L  Gtn      0  d     L  Tan 


L  Sin 


Prop.  Pts. 


75°  —  Logarithms  of  Trigonometric  Functions 


Ill]  15°  —  Logarithms  of  Trigonometric  Functions 


61 


L  Sin 


LTan 


c  d      L  Ctn 


L  Cos 


Prop.  Pts. 


9.41  300 
9.41  347 
9.41  394 
9.41  441 
9-41  488 
9.41  535 
9.41  582 
9.41  G28 
9.41  675 
9.41 722 
9.41  768 
9.41  815 
9.41  861 
9.41  908 

9.41  954 

9.42  001 
9.42  047 
9.42  093 
9.42  140 
9.42  186 
9.42  232 
9.42  278 
9.42  324 
9.42  370 
9.42  416 
9.42  461 
9.42  507 
9.42  553 
9.42  599 
9.42  644 
9.42  690 
9.42  735 
9.42  781 
9.42  826 
9.42  872 
9.42  917 

9.42  962 

9.43  008 
9.43  053 
9.43  098 
9.43 143 
9.43 188 
9.43  233 
9.43  278 
9.43  323 
9.43  367 
9.43412 
9.43  457 
9.43  502 
9.43  546 
9.43  591 
9.43  635 
9.43  680 
9.43  724 
9.43  769 
9.43  813 
9.43  857 
9.43  901 
9.43  946 
9.43  990 
9  44  034 


.42  805 
.42  856 
,42  906 
,42  957 
,43  007 
,43  057 
,43  108 
,43  158 
,43  208 
43  258 
,43  308 
43  358 
43  408 
43  458 

43  508 
.43  558 
.43  607 
.43  657 
.43  707 
.43  756 
.43  806 
.43  855 
.43  905 
.43  954 
.44  004 
.44  053 
.44  102 
.44  151 
.44  201 
.44  250 

44  299 
,44  348 
,44  397 
,44  446 
,44  495 
,44  514 
,44  592 
,44  641 
,44  690 
,44  738 
,44  787 
41  836 
,44  884 
,44  933 

44  981 
,45029 

45  078 
,45  126 
45  174 
,45  222 
45  271 
45  319 
,45  367 
45  415 
,45  463 
45  511 
,45  559 
,45  606 
45  654 
45  702 
45  750 


0.57  195 
0.57  144 
0.57  094 
0.57  043 
0.56  993 
0.56  943 
0.56  892 
0.56  842 
0.56  792 
0.56  742 
0.56  692 
0.56  642 
0.56  592 
0.56  542 
0.56  492 
0.56  442 
0.56  393 
0.56  343 
0.56  293 
0.56  244 
0.56  IM 
0.56  145 
0.56  095 
0.56  046 
0.55  996 
0.55  947 
0.55  898 
0.55  849 
0.55  799 
0.55  750 
0.55  701 
0.55  652 
0.55  603 
0.55  554 
0.55  505 
0.55  456 
0.55  408 
0.55  359 
0.55  310 
0.55  262 
0.55  213 
0.55164 
0.55116 
0.55  067 
0.55  019 
0.54  971 
0.54  922 
0.54  874 
0.54  826 
0.54  778 
0.54  729 
0.54  681 
0.54  633 
0.54  585 
0.54  537 
0.54  489 
0.54  441 
0.54  394 
0.54  346 
0.54  298 
0.54  250 


9.98  494 
9.98  491 
9.98  488 
9.98  484 
9.98  481 
9.98  477 
9.98  474 
9.98  471 
9.98  467 
9.98  464 
9.98  460 
9.98  457 
9.98  453 
9.98  450 
9.98  447 
9.98  443 
9.98  440 
9.98  436 
9-98  433 
9.98  429 
9.98.426 
9.98  422 
9.98  419 
9.98  415 
9.98  412 
9.98  409 
9.98  405 
9.98  402 
9.98  398 
9.98  395 
9.98  391 
9.98  388 
9.98  384 
9.98  381 
9.98  377 
9.98  373 
9.98  370 
9.98  366 
9.98  363 
9.98  359 
9.98  356 
9.98  352 
9.98  349 
9.98  345 
9.98  342 
9.98  338 
9-98  334 
9.98  331 
9.98  327 
9.98  324 
9.98  320 
9.98  317 
9.98  313 
9.98  309 
9.98  306 
9.98  302 
9.98  299 
9.98  295 
9.98  291 
9.98  288 
9.98  284 


51 

50 

2 

10.2 

10.0 

3 

15.3 

15.0 

4 

20.4 

20.0 

5 

25.5 

25.0 

6 

30.6 

30.0 

7 

35.7 

35.0 

8 

40.8 

40.0 

9 

45.9 

45.0 

48 

47 

2 

9.6 

.   9.4 

3 

14.4 

14.1 

4 

19.2 

18.8 

5 

24.0 

23.5 

() 

28.8 

28.2 

7 

33.6 

32.9 

8 

38.4 

37.6 

9 

43.2 

42.3 

49 

9.8 
14.7 
19.6 
24.5 
29.4 
34.3 
39.2 
44.1 


46 

9.2 
13.8 
18.4 
23.0 
27.6 
32.2 
3(1.8 
41.4 


45 

2 

9.0 

3 

13.5 

4 

18.0 

5 

22.5 

6 

27.0 

7 

31.5 

8 

36.0 

9 

40.5 

4 

2 

0.8 

3 

1.2 

4 

1.6 

5 

2.0 

6 

2.4 

7 

2.8 

8 

3.2 

9 

3.6 

44 

8.8 
13.2 
17.6 
22.0 
26.4 
30.8 
35.2 
39.6 


3 

0.6 
0.9 
1.2 
1.5 
1.8 
2.1 
2.4 
2.7 


From  the  top  : 

For  15°+  or  195°+, 
read  as  printed  ;  for 
105°+  or  285°+,  read 
co-function. 

From  the  bottom  : 

For  74°+  or  254°+, 
read  as  printed  ;  for 
164°+  or  344°+,  read 
co-function. 


L  Cos 


LCtn 


c  d   L  Tan 


L  Sin 


Prop.  Pts. 


74° — Logarithms  of  Trigonometric  Functions 


62 


16°  —  Logarithms  of  Trigonometric  Functions         [in 


L  Sin 


L  Tan     c  d      L  Ctn 


LGos 


Prop.  Pts. 


0 

1 
2 
3 
4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


9.44  034 
9.44  078 
9.44  122 
9.44  166 
9.44  210 
9.44  253 
9.44  297 
9.44  341 
9.44  385 
9.44  428 
9.44  472 
9.44  516 
9.44  559 
9.44  602 
9.44  646 
9.44  689 
9.44  733 
9.44  776 
9.44  819 
9.44  862 
9.44  905 
9.44  948 

9.44  992 

9.45  035 
9.45  077 
9.45  120 
9.45  163 
9.45  206 
9.45  249 
9.45  292 
9.45  334 
9.45  377 
9.45  419 
9.45  462 
9.45  504 
9.45  547 
9.45  589 
9.45  632 
9.45  674 
9.45  716 
9.45  758 
9.45  801 
9.45  843 
9.45  885 
9.45  927 

9.45  969 

9.46  011 
9.46053 
9.46  095 
9.46  136 
9.46 178 
9.46  220 
9.46  262 
9.46  303 
9.46  345 
9.46  386 
9.46  428 
9.46  469 
9.46  511 
9.46  552 
9.46  594 


9.45  750 
9.45  797 
9.45  845 
9.45  892 
9.45  940 

9.45  987 

9.46  035 
9.46  082 
9.46  130 
9.46  177 
9.46  224 
9.46  271 
9.46  319 
9.46  366 
9.46  413 
9.46  460 
9.46  507 
9.46  554 
9.46  601 
9.46  648 
9.46  694 
9.46  741 
9.46  788' 
9.46  835 
9.46  881 
9.46  928 

9.46  975 

9.47  021 
9.47  068 
9.47  114 
9.47  160 
9.47  207 
9.47  253 
9.47  299 
9.47  346 
9.47  392 
9.47  438 
9.47  484 
9.47  530 
9.47  576 
9.47  622 
9.47  668 
9.47  714 
9.47  760 
9.47  806 
9.47  852 
9.47  897 
9.47  943 

9.47  989 

9.48  035 
9.48080 
9.48  126 
9.48  171 
9.48  217 
9.48  262 
9.48  307 
9.48  353 
9.48  398 
9.48  443 
9.48  489 
9.48  534 


0.54  250 
0.54  203 
0.54  155 
0.54  108 
0.64  060 
0.54  013 
0.53  9(i5 
0.53  918 
0.53  870 
0.53  823 
0.53  776 
0.53  729 
0.53  681 
0.53  634 
0.53  587 
0.53  540 
0.53  493 
0.53  446 
0.53  399 
0.53  352 
0.53  306 
0.53  259 
0.53  212 
0.53  165 
0.53  119 
0.53072 
0.53  025 
0.52  979 
0.52  932 
0.52  886 
0.52  840 
0.52  793 
0.52  747 
0.52  701 
0.52  654 
0.52  608 
0.52  562 
0.52  516 
0.52  470 
0.52  424 
0.52  378 
0.52  332 
0.52  286 
0.52  240 
0.52  194 
0.52 148 
0.52  103 
0.52  057 
0.52  011 
0.51 965 
0.51 920 
0.51  874 
0.51  829 
0.51  783 
0.51  738 
0.51693 
0.51 647 
0.51 602 
0.51  557 
0.51 511 
0.51  466 


9.98  284 
9.98  281 
9.98  277 
9.98  273 
9.98  270 
9.98  266 
9.98  262 
9.98  259 
9.98  255 
9.98  251 
9.98  248 
9.98  244 
9.98  240 
9.98  237 
9.98  233 
9.98  229 
9.98  226 
9.98  222 
9.98  218 
9.98  215 
9.98  211 
9.98  207 
9.98  204 
9.98  200 
9.98  196 
9.98  192 
9.98  189 
9.98  185 
9.98  181 
9.98  177 
9.98 174 
9.98 170 
9.98  166 
9.98  162 
9.98  159 
9.98  155 
9.98 151 
9.98  147 
9.98  144 
9.98  140 
9.98  136 
9.98  132 
9.98  129 
9.98  125 
9.98 121 
9.98 117 
9.98  113 
9.98 110 
9.98  106 
9.98  102 
9.98  098 
9.98094 
9.98  090 
9.98  087 
9.98  083 
9.98  079 
9.98  075 
9.98071 
9.98  067 
9.98063 
9.98  060 


48 

47 

2 

9.6 

9.4 

3 

14.4 

14.1 

4 

19.2 

18.8 

5 

24.0 

23.5 

6 

28.8 

28.2 

7 

33.6 

32.9 

8 

38.4 

37.6 

9 

43.2 

42.3 

45 

44 

2 

9.0 

8.8 

3 

13.5 

13.2 

4 

18.0 

17.6 

5 

22.5 

22.0 

6 

27.0 

26.4 

7 

31.5 

30.8 

8 

36.0 

35.2 

9 

40.5 

39.6 

46 

9.2 
13.8 
18.4 
23.0 

27.6 
32.2 
36.8 
41.4 

43 

8.6 
12.9 
17.2 
21.5 

25.8 
30.1 
34.4 
38.7 


42 

2 

8.4 

3 

12.6 

4 

16.8 

5 

21.0 

6 

25.2 

7 

29.4 

8 

33.6 

9 

37.8 

4 

2 

0.8 

3 

1.2 

4 

1.6 

5 

2.0 

6 

2.4 

7 

2.8 

8 

3.2 

9 

3.6 

41 

8.2 
12.3 
16.4 
20.5 
24.6 
28.7 
32.8 
36.9 


0.6 
0.9 
1.2 
1.5 
1.8 
2.1 
2.4 
2.7 


From  the  top  : 

For  16°+ or  196°+, 
read  as  printed;  for 
106°+  or  286°+,  read 
co-function. 

From  the  bottom : 

For  73°+  or  253°+, 
read  as  printed  ;  for 
163°+  or  343°+,  read 
co-function. 


LGos 


L  Ctn    led     L  Tan 


L  Sin      d     ' 


Prop.  Pts. 


73°— Logarithms  of  Trigononietric  Functions 


m]  17°  —  Logarithms  of  Trigonometric  Functions 


63 


L  Sin 


LTan 


c  d     L  Ctn 


L  Cos 


Prop.  Pts. 


9.46  594 
9.46  635 
9.4(3  676 
9.46  717 
9.46  758 
9.46  800 
9.46  841 
9.46  882 
9.46  923 

9.46  9(54 

9.47  005 
9.47  045 
9.47  086 
9.47  127 
9.47  168 
9.47  209 
9.47  249 
9.47  290 
9.47  330 
9.47  371 
9.47  411 
9.47  452 
9.47  492 
9.47  533 
9.47  573 
9.47  613 
9.47  654 
9.47  694 
9.47  734 
9.47  774 
9.47  814 
9.47  854 
9.47  894 
9.47  934 

9.47  974 

9.48  014 
9.48  054 
9.48  094 
9.48  133 
9.48  173 
9.48  213 
9.48  252 
9.48  292 
9.48  332 
9.48  371 
9.48  411 
9.48  450 
9.48  490 
9.48  529 
9.48  568 
9.48  607 
9.48  (i47 
9.48  686 
9.48  725 
9.48  764 
9.48  803 
9.48  842 
9.48  881 
9.48  920 
9.48  959 
9.48  998 


9.48  534 
9.48  579 
9.48  ()24 
9.48  669 
9.48  714 
9.48  759 
9.48  804 
9.48  849 
9.48  894 
9.48  939 

9.48  984 

9.49  029 
9.49  073 
9.49  118 
9.49  163 
9.49  207 
9.49  252 
9.49  296 
9.49  341 
9.49  385 
9.49  430 
9.49474 
9.49  519 
9.49  563 
9.49  607 
9.49  652 
9.49  696 
9.49  740 
9.49  784 
9.49  828 
9.49  872 
9.49  916 

9.49  960 

9.50  004 
9.50048 
9.50  092 
9.50  136 
9.50  180 
9.50  223 
9.50  267 
9.50  311 
9.50  355 
9.50  398 
9.50  442 
9.50485 
9.50  529 
9.50  572 
9.50  616 
9.50  659 
9.50  703 
9.50  746 
9.50  789 
9.50  833 
9.50  876 
9.50  919 

9.50  962 
9.51 005 

9.51  048 
9.51 092 
9.51 135 
9.51 178 


0.51  4(56 
0.51 421 
0.51  376 
0.51  331 
0.51  286 
0.51 241 
0.51 196 
0.51 151 
0.51 106 
0.51  061 
0.51  016 
0.50  971 
0.50  927 
0.50  882 
0.50  837 
0.50  793 
0.50  748 
0.50  704 
0.50  659 
0.50  615 
0.50  570 
0.50  526 
0.50  481 
0.50  437 
0.50  393 
0.50  348 
0.50  304 
0.50  260 
0.50  216 
0.50  172 
0.50  128 
0.50  084 
0.50  040 
0.49  996 
0.49  952 
0.49  908 
0.49  864 
0.49  820 
0.49  777 
0.49  733 
0.49  689 
0.49  645 
0.49  602 
0.49  558 
0.49  515 
0.49  471 
0.49  428 
0.49  384 
0.49  341 
0.49  297 
0.49  254 
0.49  211 
0.49  167 
0.49 124 
0.49  081 
0.49  038 
0.48  995 
0.48  952 
0.48  908 
0.48  865 
0.48  822 


9.98  060 
9.98  056 
9.98  052 
9.98  048 
9.98  044 
9.98  040 
9.98  036 
9.98  032 
9.98  029 
9.98  025 
9.98  021 
9.98  017 
9.98  013 
9.98  009 
9.98  005 
9.98  001 
9.97  997 
9.97  993 
9.97  989 
9.97  986 
9.97  982 
9.97  978 
9.97  974 
9.97  970 
9.97  966 
9.97  962 
9.97  958 
9.97  954 
9.97  950 
9.97  946 
9.97  942 
9.97  938 
9.97  934 
9.97  930 
9.97  926 
9.97  922 
9.97  918 
9.97  914 
9.97  910 
9.97  906 
9.97  902 
9.97  898 
9.97  894 
9.97  890 
9.97  886 
9.97  882 
9.97  878 
9.97  874 
9.97  870 
9.97  866 
9.97  861 
9.97  857 
9.97  853 
9.97  849 
9.97  845 
9.97  841 
9.97  837 
9.97  833 
9.97  829 
9.97.825 
9.97  821 


45 

44 

2 

9.0 

8.8 

3 

13.5 

13.2 

4 

18.0 

17.6 

5 

22.5 

22.0 

6 

27.0 

26.4 

7 

31.5 

30.8 

8 

36.0 

35.2 

9 

40.5 

39.6 

42 

41 

2 

8.4 

8.2 

3 

12.6 

12.3 

4 

16.8 

16.4 

5 

21.0 

20.5 

6 

25.2 

24.6 

7 

29.4 

28.7 

8 

33.6 

32.8 

9 

37.8 

36.9 

43 

8.6 
12.9 
17.2 
21.5 
25.8 
30.1 
34.4 
38.7 

40 

8.0 
12.0 
16.0 
20.0 
24.0 
28.0 
32.0 
36.0 


39 

2 

7.8 

3 

11.7 

4 

15.6 

5 

19.5 

6 

23.4 

7 

27.3 

8 

31.2 

9 

35.1 

2 

4 

0.8 

3 

1.2 

4 

1.6 

5 

2.0 

6 

2.4 

7 

2.8 

8 

3.2 

9 

3.6 

1.0 
1.5 
2.0 
2.5 
3.0 
3.5 
4.0 
4.5 


3 

0.6 
0.9 
1.2 
1.5 
1.8 
2.1 
2.4 
2.7 


From  the  top : 

For  17°+  or  197^+, 
read  as  printed  ;  for 
107°+ or  287°+,  read 
co-function . 

From  the  bottom: 

For  72°+  or  252°+, 
read  as  printed  ;  for 
162°+or  342°+,  read 

co-function. 


L  Cos 


LCtn 


c  d  L  Tan 


L  Sin 


d  ' 


Prop.  Pts. 


72° — Logarithms  of  Trigonometric  Functions 


64 


18°  — Logarithms  of  Trigonometric  Tunctions 


L  Sin 


L  Tan 


c  d      L  Ctn 


L  Cos 


Prop.  Pts. 


0 

1 
2 
3 
4 
6 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
•39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


9.48  998 

9.49  037 
9.49076 
9.49 115 
9.49 153 
9.49 192 
9.49  231 
9.49  269 
9.49  308 
9.49  347 
9.49  385 
9.49  424 
9.49462 
9.49  500 
9.49  539 
9.49  577 
9.49615 
9.49654 
9.49  692 
9.49  730 
9.49  768 
9.49806 
9.49  844 
9.49  882 
9.49  920 
9.49  958 

9.49  996 

9.50  034 
9.50  072 
9.50 110 
9.50  148 
9.50 185 
9.50  223 
9.50  261 
9.50  298 
9.50  3.36 
9.50  374 
9.50  411 
9.50  449 
9.50  486 
9.50  523 
9.50  561 
9.50  598 
9.50  635 
9.50  673 
9.50  710 
9.50  747 
9.50  784 
9.50  821 
9.50  858 
9.50  896 
9.50  933 

9.50  970 
9.51007 

9.51  043 
9.51 080 
9.51 117 
9.51 154 
9.51 191 
9.51  227 
9.51  264 


9.51 178 
9.51  221 
9.51  264 
9.51  306 
9.51  349 
9.51  392 
9.51  435 
9.51  478 
9.51  520 
9.51  563 
9.51  606 
9.51  648 
9.51  691 
9.51  734 
9.51  776 
9.51  819 
9.51  861 
9.51  903 
9.51  946 

9.51  988 

9.52  031 
9.52  073 
9.52  115 
9.52  157 
9.52  200 


52  242 
52  284 
52  326 
52  368 
52  410 
,52  452 
52  494 
52  536 
.52  578 
52  620 
52  661 
52  703 
52  745 
52  787 
,52  829 
52  870 
52  912 
52  953 

52  995 

53  037 
,53  078 
53120 
53 161 
53  202 
53  244 
,53  285 
,53  327 
53  368 
53  409 
53  450 
,53  492 
,53  533 
,53  574 
53  615 
,53  656 
,53  697 


0.48  822 
0.48  779 
0.48  736 
0.48  694 
0.48  651 
0.48  608 
0.48  565 
0.48  522 
0.48  480 
0.48  437 
0.48  394 
0.48  352 
0.48  309 
0.48  266 
0.48  224 
0.48  181 
0.48  139 
0.48  097 
0.48  054 
0.48  012 
0.47  ^69 
0.47  927 
0.47  885 
0.47  843 
0.47  800 
0.47  758 
0.47  716 
0.47  674 
0.47  632 
0.47  590. 
0.47  548 
0.47  506 
0.47  464 
0.47  422 
0.47  380 
0.47  339 
0.47  297 
0.47  255 
0.47  213 
0.47  171 
0.47  130 
0.47  088 
0.47  047 
0.47  005 
0.46  963 
0.46  922 
0.46  880 
0.46  839 
0.46  798 
0.46  756 
0.46  715 
0.46  673 
0.46  632 
0.46  591 
0.46  550 
0.46  508 
0.46  467 
0.46  426 
0.46  385 
0.46  344 
0.46  303 


9.97  821 
9.97  817 
9.97  812 
9.97  808 
9.97  804 
9.97  800 
9.97  796 
9.97  792 
9.97  788 
9.97  784 
9.97  779 
9.97  775 
9.97  771 
9.97  767 
9.97  763 
9.97  759 
9.97  754 
9.97  750 
9.97  746 
9.97  742 
9.97  738 
9.97  734 
9.97  729 
9.97  725 
9.97  721 
9.97  717 
9.97  713 
9.97  708 
9.97  704 
9.97  700 
9.97  696 
9.97  691 
9.97  687 
9.97  683 
9.97  679 
9.97  674 
9.97  670 
9.97  666 
9.97  662 
9.97  657 
9.97  653 
9.97  649 
9.97  645 
9.97  ()40 
9.97  636 
9.97  632 
9.97  628 
9.97  623 
9.97  619 
9.97  615 
9.97  610 
9.97  606 
9.97  602 
9.97  597 
9.97  593 
9.97  589 
9.97  584 
9.97  580 
9.97  576 
9.97  571 
9.97  567 


43 

42 

2 

8.6 

8.4 

3 

12.9 

12.6 

4 

17.2 

16.8 

5 

21.5 

21.0 

6 

25.8 

25.2 

7 

30.1 

29.4 

8 

34.4 

33.6 

9 

38.7 

37.8 

39 

38 

2 

7.8 

7.6 

3 

.11.7 

11.4 

4 

15.6 

15.2 

5 

19.5 

19.0 

6 

23.4 

22.8 

7 

27.3 

26.6 

8 

31.2 

30.4 

9 

35.1 

34.2 

36 

5 

2 

7.2 

1.0 

3 

10.8 

1.5 

4 

14.4 

2.0 

5 

18.0 

2.5 

6 

21.6 

3.0 

7 

25.2 

3.5 

8 

28.8 

4.0 

9 

32.4 

4.5 

41 

8.2 
12.3 
16.4 
20.5 
24.6 
28.7 
32.8 
36.9 


37 

7.4 
11.1 
14.8 
18.5 
22.2 
25.9 
29.6 
33.3 


0.8 
1.2 
1.6 
2.0 
2.4 
2.8 
3.2 
3.6 


From  the  top  : 

For  18°+ or  198°+, 

read  as  printed  ;  for 
108°+  or  288°+,  read 
co-function. 

From  the  bottom : 

For  71°+ or  251°+, 
read  as  printed;  for 
161°+  or  341°+,  read 
co-function. 


LGos 


LCtn 


0  d  L  Tan 


L  Sin 


Prop.  Pts. 


7r  —  Logarithms  of  Trigonometric  Functions 


Ill] 


19°  —  Logarithms  of  Trigonometric  Functions 


65 


LSin 


L  Tan 


c  d      L  Ctn 


L  Cos 


Prop.  Pts. 


0 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 

45 

46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
'57 
58 
59 
60 


9.51  264 
9.51 301 
9.51  338 
9.51  374 
9.51  411 
9.51 447 
9.51  484 
9.51  520 
9.51  557 
9.51 593 
9.51 629 
9.51  666 
9.51  702 
9.51  738 
9.51774 
9.51811 
9.51  847 
9.51  883 
9.51  919 
9.51  955 

9.51  991 

9.52  027 
9.52  063 
9.52  099 
9.52  135 
9.52  171 
9.52  207 
9.52  242 
9.52  278 
9.52  314 
9.52  350 
9.52  385 
9.52  421 
9.52  456 
9.52  492 
9.52  527 
9.52  563 
9.52  598 
9.52  634 
9.52  669 
9.52  705 
9.52  740 
9.52  775 
9.52  811 
9.52  846 
9.52  881 
9.52  916 
9.52  951 

9.52  986 

9.53  021 
9.53  056 
9.53092 
9.53  126 
9.53  161 
9.53  196 
9.53  231 
9.53  266 
9.53  301 
9.53  336 
9.53  370 
9.53405 


9.53  697 
9.53  738 
9.53  779 
9.53  820 
9.53  861 
9.53  902 
9.53  943 

9.53  984 

9.54  025 
9.54  065 
9.54 106 
9.54  147 
9.54  187 
9.54  228 
9.54  269 
9.54  309 
9.54  350 
9.54  390 
9.54  431 
9.54  471 
9.54  512 
9.54  552 
9.54  593 
9.54  633 
9.54  673 
9.54  714 
9.54  754 
9.54  794 
9.54  835 
9.54  875 
9.54  915 
9.54  955 

9.54  995 

9.55  035 
9.55  075 
9.55  115 
9.55  155 
9.55  195 
9.55  235 
9.55  275 
9.55  315 
9.55  355 
9.55  395 
9.55  434 
9.55  474 
9.55  514 
9.55  554 
9.55  593 
9.55  633 
9.55  673 
9.55  712 
9.55  752 
9.55  791 
9.55  831 
9.55  870 
9.55  910 
9.55  949 

9.55  989 

9.56  028 
9.56  067 
9.56  107 


0.46  303 
0.46  262 
0.46  221 
0.46  180 
0.46  139 
0.46  098 
0.46  057 
0.46  016 
0.45  975 
0.45  935 
0.45  894 
0.45  853 
0.45  813 
0.45  772 
0.45  731 
0.45  691 
0.45  650 
0.45  610 
0.45  569 
0.45  529 
0.45  488 
0.45  448 
0.45  407 
0.45  367 
0.45  327 
0.45  286 
0.45  246 
0.45  206 
0.45  165 
0.45  125 
0.45  085 
0.45  045 
0.45  005 
0.44  965 
0.44  925 
0.44  885 
0.44  845 
0.44  805 
0.44  765 
0.44  725 
0.44  685 
0.44  645 
0.44  605 
0.44  566 
0.44  526 
0.44  486 
0.44  446 
0.44  407 
0.44  367 
0.44  327 
0.44  288 
0.44  248 
0.44  209 
0.44 169 
0.44  130 
0.44090 
0.44  051 
0.44  011 
0.43  972 
0.43  933 
0.43  893 


9.97  567 
9.97  563 
9.97  558 
9.97  554 
9.97  550 
9.97  545 
9.97  541 
9.97  536 
9.97  532 
9.97  528 
9.97  523 
9.97  519 
9.97  515 
9.97  510 
9.97  506 
9.97  501 
9.97  497 
9.97  492 
9.97  488 
9.97  484 
9.97  479 
9.97  475 
9.97  470 
9.97  466 
9.97  461 
9.97  457 
9.97  453 
9.97  448 
9.97  444 
9.97  439 
9.97  435 
9.97  430 
9.97  426 
9.97  421 
9.97  417 
9.97  412 
9.97  408 
9.97  403 
9.97  399 
9.97  394 
9.97  390 
9.97  385 
9.97  381 
9.97  376 
9.97  372 
9.97  367 
9.97  363 
9.97  358 
9.97  353 
9.97  349 
9.97  344 
9.97  340 
9.97  335 
9.97  331 
9.97  326 
9.97  322 
9.97  317 
9.97  312 
9.97  308 
9.97  303 
9.97  299 


41 

40 

2 

8.2 

8.0 

3 

12.3 

12.0 

4 

16.4 

16.0 

5 

20.5 

20.0 

6 

24.6 

24.0 

7 

28.7 

28.0 

8 

32.8 

32.0 

9 

36.9 

36.0 

37 

36 

2 

7.4 

7.2 

3 

11.1 

10.8 

4 

14.8 

14.4 

5 

18.5 

18.0 

6 

22.2 

21.6 

7 

25.9 

25.2 

8 

29.6 

28.8 

9 

33.3 

32.4 

2 

34 

6.8 

5 

1.0 

3 

10.2 

1.5 

4 

13.6 

2.0 

5 

17.0 

2.5 

6 

20.4 

3.0 

7 

23.8 

3.5 

8 

27.2 

4.0 

9 

30.6 

4.5 

39 

7.8 
11.7 
15.6 
19.5 
23.4 
27.3 
31.2 
35.1 


35 

7.0 
10.5 
14.0 
17.5 

21.0 
24.5 
28.0 
31.5 


0.8 
1.2 
1.6 
2.0 
2.4 
2.8 
3.2 
3.6 


From  the  top : 

For  19°+  or  199°+, 

read  as  printed  ;  for 
109°+  or  289°+,  read 
co-functioD. 

From  the  bottom : 

For  70°+  or  250°+, 

read  as  printed  ;  for 
160°+  or  340°+,  read 
co-function. 


LCos 


LCtn 


0  d   L  Tan 


L  Sin 


Prop.  Pts. 


70°— Logarithms  of  Trigonometric  Functions 


66 


20°  —  Logarithms  of  Trigonometric  Functions         [in 


LSin 


L  Tan     c  d      L  Ctn 


LGos 


Prop.  Pts. 


0 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


9.53  405 
9.53  440 
9.53  475 
9.53  509 
9.53  544 
9.53  578 
9.53  613 
9.53  647 
9.53  682 
•9.53  716 
9.53  751 
9.53  785 
9.53  819 
9.53  854 
9.53  888 
9.53  922 
9.53  957 

9.53  991 

9.54  025 
9.54059 
9  54  093 
9.54  127 
9.54  161 
9.54  195 
9.54  229 
9.54  263 
9.54  297 
9.54  331 
9.54  365 
9.54  399 
9.54  433 
9.54  466 
9.54  500 
9.54  534 
9.54  567 
9.54  601 
9.54  635 
9.54  668 
9.54  702 
9.54  735 
9.54  769 
9.54  802 
9.54  836 
9.54  869 
9.54  903 
9.54  936 

9.54  969 

9.55  003 
9.55  036 
9.55  069 
9.55  102 
9.55 136 
9.55  169 
9.55  202 
9.55  235 
9.55  268 
9.55  301 
9.55  334 
9.55  367 
9.55  400 
9.55  433 


9.56  107 
9.56  146 
9.56  185 
9.56  224 
9.56  264 
9.56  303 
9.56  342 
9.56  381 
9.56  420 
9.56  459 
9.56  498 
9.56  537 
9.56  576 
9.56  615 
9.56  654 
9.56  693 
9.56  732 
9.56  771 
9.56  810 
9.56  849 
9.56  887 
9.56  926 

9.56  965 

9.57  004 
9.57  042 
9.57  081 
9.57  120 
9.57  158 
9.57  197 
9.57  235 
9.57  274 
9.57  312 
9.57  351 
9.57  389 
9.57  428 
9.57  466 
9.57  504 
9.57  543 
9.57  581 
9.57  619 
9.57  658 
9.57  696 
9.57  734 
9.57  772 
9.57  810 
9.57  849 
9.57  887 
9.57  925 

9.57  963 

9.58  001 
9.58  039 
9.58  077 
9.58  115 
9.58  153 
9.58  191 
9.58  229 
9.58  267 
9.58  304 
9.58  342 
9.58  380 
9.58  418 


43  893 

43  854 

43  815 

43  776 

43  736 

43  697 

43  658 

43  619 

43  580 

43  541 

43  502 

43  463 

43  424 

43  385 

43  346 

0.43  307 

0.43  268 

0.43  229 

0.43  190 

0.43  151 

0.43  113 

0.43  074 

0.43  035 

0.42  996 

0.42  958 

0.42  919 

0.42  880 

0.42  842 

0.42  803 

0.42  765 

0.42  726 

0.42  688 

0.42  649 

0.42  611 

0.42  572 

0,42  534 

0.42  496 

0.42  457 

0.42  419 

0.42  381 

0.42  342 

0.42  304 

0.42  266 

0.42  228 

0.42  190 

0.42  151 

0.42  113 

0.42  075 

0.42  037 

0.41  999 

0.41  961 

0.41  923 

0.41  885 

0.41  847 

0.41  809 

0.41  771 

0.41  733 

0.41 696 

0.41  658 

0.41 620 

0.41  582 


9.97  299 
9.97  294 
9.97  289 
9.97  285 
9.97  280 
9.97  276 
9.97  271 
9.97  266 
9.97  262 
9.97  257 
9.97  252 
9.97  248 
9.97  243 
9.97  238 
9.97  234 
9.97  229 
9.97  224 
9.97  220 
9.97  215 
9.97  210 
9.97  206 
9.97  201 
9.97  196 
9.97  192 
9.97  187 
9.97  182 
9.97  178 
9.97  173 
9.97  168 
9.97  163 
9.97  159 
9.97  154 
9.97  149 
9.97  145 
9.97  140 
9.97  135 
9.97  130 
9.97  126 
9.97  121 
9.97  116 
9.97  111 
9.97  107 
9.97  102 
9.97  097 
9.97  092 
9.97  087 
9.97  083 
9.97  078 
9.97  073 
9.97  068 
9.97  063 
9.97  059 
9.97  054 
9.97  049 
9.97  044 
9.97  039 
9.97  035 
9.97  030 
9.97  025 
9.97  020 
9.97  015 


40 

39 

2 

8.0 

7.8 

3 

12.0 

11.7 

4 

16.0 

15.6 

5 

20.0 

19.5 

6 

24.0 

23.4 

7 

28.0 

27.3 

8 

32.0 

31.2 

9 

36.0 

35.1 

37 

35 

2 

7.4 

7.0 

3 

11.1 

10.5 

4 

14.8 

14.0 

5 

18.5 

17.5 

6 

22.2 

21.0 

7 

25.9 

24.5 

8 

29.6 

28.0 

9 

33.3 

31.5 

33 

5 

2 

6:6 

1.0 

3 

9.9 

1.5 

4 

13.2 

2.0 

5 

16.5 

2.5 

6 

19.8 

3.0 

7 

23.1 

3.5 

8 

26.4 

4.0 

9 

29.7 

4.5 

From  the  top : 

For  20°+  or  200°+, 

read  as  printed;  for 
110°+  or  290°+,  read 
co-function. 

From  the  bottom : 

For  69°+  or  249°+, 
read  as  printed  ;  for 
159°+  or  339°+, read 
co-function. 


LCos 


LGtn 


0  d     L  Tan 


LSin 


Prop.  Pts. 


fi9° 


-TiOo*a,ritliTns  nf  Trie^ononnpifrin  Fiinr»tioTis 


Ill] 


21°  —  Logarithms  of  Trigonometric  Functions 


67 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 


L  Sin 


9.55  433 
9.55  46() 
9.55  499 
9.55  532 
9.55  564 
9.55  597 
9.55  630 
9.55  663 
9.55  695 
9.55  728 
9.55  761 
9.55  793 
9.55  826 
9.55  858 
9.55  891 
9.55  923 
9.55  956 

9.55  988 

9.56  021 
9.56  053 
9.56  085 
9.56  118 
9.56  150 
9.56  182 
9.56  215 
9.56  247 
9.56  279 
9.56  311 
9.56  343 
9.56  375 
9.56  408 
9.56  440 
9.56  472 
9.56  504 
9.56  536 
9.56  568 
9.56  599 
9.56  631 
9.56  663 
9.56  695 
9.56  727 
9.56  759 
9.56  790 
9.56  822 
9.56  854 
9.56  886 
9.56  917 
9.56  949 

9.56  980 

9.57  012 
9.57  044 
9.57  075 
9.57  107 
9.57  138 
9.57  169 
9.57  201 
9.57  232 
9.57  264 
9.57  295 
9.57  326 
9.57  358 


L  Tan  c  d  L  Ctn 


9.58  418 
9.58  455 
9.58  493 
9.58  531 
9.58  569 
9.58  606 
9.58  644 
9.58  681 
9.58  719 
9.58  757 
9.58  794 
9.58  832 
9.58  869 
9.58  907 
9.58  944 

9.58  981 
9.59019 
9.59056 
9  59  094 

9.59  131 
9.59 168 
9.59  205 
9.59  243 
9.59  280 
9.59  317 
9.59  354 
9.59  391 
9.59  429 
9.59  466 
9.59  503 
9.59  540 
9.59  577 
9.59  614 
9.59  651 
9.59  688 
9.59  725 
9.59  762 
9.59  799 
9.59  835 
9.59  872 
9.59  909 
9.59  946 

9.59  983 

9.60  019 
9.60  056 
9.60  093 
9.60  130 
9.60  166 
9.60  203 
9.60  240 
9.60  276 
9.60  313 
9.60  349 
9.60  386 
9.60  422 
9.60  459 
9.60  495 
9.60  532 
9.60  568 
9.60  605 
9.60  641 


0.41 582 
0.41  545 
0.41  507 
0.41  469 
0.41  431 
0.41  394 
0.41  356 
0.41  319 
0.41  281 
0.41  243 
0.41  206 
0.41 168 
0.41 131 
0.41  093 
0.41  056 
0.41019 
0.40  981 
0.40  944 
0.40  906 
0.40  869 
0.40  832 
0.40  795 
0.40  757 
0.40  720 
0.40  683 
0.40  646 
0.40  609 
0.40  571 
0.40  534 
0.40  497 
0.40  460 
0.40  423 
0.40  386 
0.40  349 
0.40  312 
0.40  275 
0.40  238 
0.40  201 
0.40  165 
0.40  128 
0.40091 
0.40  054 
0.40  017 
0.39  981 
0.39  944 
0.39  907 
0.39  870 
0.39  834 
0.39  797 
0.39  760 
0.39  724 
0.39  687 
0.39  651 
0.39  614 
0.39  578 
0.39  541 
0.39  505 
0.39  468 
0.39  432 
0.39  395 
0.39  359 


L  Cos 


9.97  015 
9.97  010 
9.97  005 
9.97  001 
9.96  996 
9.96  991 
9.96  986 
9.96  981 
9.96  976 
9.96  971 
9.96  966 
9.96  962 
9.96  957 
9.96  952 
9.96  947 
9.96  942 
9.96  937 
9.96  932 
9.96  927 
9.96  922 
9.96  917 
9.96  912 
9.96  907 
9.96  903 
9.96  898 
9.96  893 
9.96  888 
9.f)6  883 
9.96  878 
9.96  873 
9.96  868 
9.96  863 
9.96  858 
9.96  853 
9.96  848 
9.96  843 
9.96  838 
9.96  833 
9.96  828 
9.96  823 
9.96  818 
9.96  813 
9.96  808 
9.96  803 
9.96  798 
9.96  793 
9.96  788 
9.96  783 
9.96  778 
9.96  772 
9.96  767 
9.96  762 
9.96  757 
9.96  752 
9.96  747 
9.96  742 
9.96  737 
9.96  732 
9.96  727 
9.96  722 
9.96  717 


Prop.  Pts. 


38 

37 

2 

7.6 

7.4 

3 

11.4 

11.1 

4 

15.2 

14.8 

5 

19.0 

18.5 

6 

22.8 

22.2 

7 

26.6 

25.9 

8 

30.4 

29.6 

9 

34.2 

33.3 

33 

32 

2 

6.6 

6.4 

3 

9.9 

9.6 

4 

13.2 

12.8 

5 

16.5 

16.0 

6 

19.8 

19.2 

7 

23.1 

22.4 

8 

26.4 

25.6 

9 

29.7 

28.8 

6 

5 

2 

1.2 

1.0 

3 

1.8 

1.5 

4 

2.4 

2.0 

5 

3.0 

2.5 

6 

3.6 

3.0 

7 

4.2 

3.5 

8 

4.8 

4.0 

9 

5.4 

4.5 

36 

7.2 
10.8 
14.4 
18.0 
21.6 
25.2 
28.8 
32.4 


31 

6.2 
9.3 
12.4 
15.5 
18.6 
21.7 
24.8 
27.9 


0.8 
1.2 
1.6 
2.0 
2.4 
2.8 
3.2 
3.6 


From  the  top: 

For  21°+  or  201°+, 

read  as  printed  ;  for 
^11°+ or  291°+, read 
co-function. 

From  the  bottom: 

For  68°+  or  248°+, 

read  as  printed  ;  for 
158°+ or  338°+,  read 

co-function. 


LGos 


L  Ctn     c  d     L  Tan 


L  Sin 


Prop.  Pts. 


68°— Logarithms  of  Trigonometric  Functions 


68 


22°  — Logarithms  of  Trigonometric  Functions 


[HI 


L  Sin 


L  Tan     c  d      L  Ctn 


L  Cos 


Prop.  Pts. 


0 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


9.57  358 
9.57  389 
9.57  420 
9.57  451 
9.57  482 
9.57  514 
9  57  545 
9.57  576 
9.57  607 
9.57  638 
9.57  669 
9.57  700 
9.57  731 
9.57  762 
9.57  793 
9.57  824 
9.57  855 
9.57  885 
9.57  916 
9.57  947 

9.57  978 

9.58  008 
9.58  039 
9.58  070 
9.58  101 
9.58  131 
9.58 162 
9.58  192 
9.58  223 
9.58  253 
9.58  284 
9.58  314 
9.58  345 
9.58  375 
9.58  406 
9.58  436 
9.58  467 
9.58  497 
9.58  527 
9.58  557 
9.58  588 
9.58  618 
9.58  648 
9.58  678 
9.58  709 
9.58739 
9.58  769 
9.58  799 
9.58  829 
9.58  859 
9.58  889 
9.58  919 
9  58  949 

9.58  979 

9.59  009 
9  59  039 
9.59069 
9.59  098 
9.59128 
9.59  158 
9.59  188 


9.60'641 
9.60  677 
9.60  714 
9.60  750 
9.60  786 
9.60  823 
9.60  859 
9.60  895 
9.60  931 

9.60  967 

9.61  004 
9.61  040 
9.61  076 
9.61 112 
9.61 148 
9.61 184 
9.61  220 
9.61  256 
9.61  292 
9.61  328 
9.61  364 
9.61  400 
9.61  436 
9.61  472 
9.61  508 
9.61  544 
9.61  579 
9.61  615 
9.61  651 
9.61  687 
9.61  722 
9.61  758 
9.61  794 
9.61  830 
9.61  865 
9.61  901 
9.61  936 

9.61  972 

9.62  008 
9.62  043 
9.62  079 
9.62  114 
9.62  150 
9.62  185 
9.62  221 
9.62  256 
9.62  292 
9.62  327 
9.62  362 
9.62  398 
9.62  433 
9.62  468 
9.62  504 
9.62  539 
9.62  574 
9.62  609 
9.62  645 
9.62  680 
9.62  715 
9.62  750 
9.62  785 


0.39  359 
0.39  323 
0.39  286 
0.39  250 
0.39  214 
0.39 177 
0.39  141 
0.39  105 
0.39  069 
0.39  0.J3 
0.38  996 
0.38  960 
0.38  924 
0.38  888 
0.38  852 
0..38  816 
0.38  780 
0.38  744 
0.38  708 
0.38  672 
0.38  636 
0.38  600 
0.38  564 
0.38  528 
0.38  492 
0.38  456 
0.38  421 
0.38  385 
0.38  349 
0.38  313 
0.38  278 
0.38  242 
0.38  206 
0.38  170 
0.38  135 
0.38  099 
0.38  064 
0.38  028 
0.37  992 
0.37  957 
0.37  921 
0.37  886 
0.37  850 
0.37  815 
0.37  779 
0.37  744 
0.37  708 
0  37  673 
0.37  638 
0.37  602 
0.37  567 
0.37  532 
0.37  496 
0.37  461 
0.37  426 
0.37  391 
0.37  355 
0.37  320 
0.37  285 
0.37  250 
0.37  215 


9.96  717 
9.96  711 
9.96  706 
9.96  701 
9.96  696 
9.96  691 
9.96  686 
9.96  681 
9.96  676 
9.96  670 
9.96  665 
9.96  660 
9.96  655 
9.96  650 
9.96  645 
9.96  640 
9.96  634 
9.96  629 
9.96  624 
9.96  619 
9.96  614 
9.96  608 
9.96  603 
9.96  598 
9.96  593 
9.96  588 
9.96  582 
9.96  577 
9.96  572 
9.96  567 
9.96  562 
9.96  556 
9.96  551 
9.96  546 
9.96  541 
9.96  535 
9.96  530 
9.96  525 
9.96  520 
9.96  514 
9.96  509 
9.96  504 
9.96  498 
9.96  493 
9.96  488 
9.96  483 
9.96  477 
9.96  472 
9.96  467 
9.96  461 
9.96  456 
9.96  451 
9.96  445 
9.96  440 
9.96435 
9.96  429 
9.96  424 
9.96  419 
9.96  413 
9.96  408 
9.96403 


5 

33 

5 

32 

5 

31 

f] 

30 

t^ 

29 

5 

28 

5 

27 

6 

26 

5 

25 

<y 

24 

»1 

23 

e, 

22 

5 

21 

,5 

20 

6 

19 

f> 

18 

.5 

17 

5 

16 

37 

86 

2 

7.4 

7.2 

3 

11.1 

10.8 

4 

14.8 

14.4 

5 

18.5 

18.0 

6 

22.2 

21.6 

7 

25.9 

25.2 

8 

29.6 

28.8 

9 

33.3 

32.4 

32 

31 

2 

6.4 

6.2 

3 

9.6 

9.3 

4 

12.8 

12.4 

5 

16.0 

15.5 

6 

19.2 

18.6 

7 

22.4 

21.7 

8 

25.6 

24.8 

9 

28.8 

27.9 

35 

7.0 
10.5 
14.0 
17.5 
21.0 
24.5 
28.0 
31.5 


30 

6.0 
9.0 
12.0 
15.0 
18.0 
21.0 
24.0 
27.0 


5 

1.0 
1.5 
2.0 
2.5 
3.0 
3.5 
4.0 
4.5 


From  the  top : 

For  22°+  or  202°+, 

read  as  printed  ;  for 
112°+  or  292°+,  read 
co-function. 

From  the  bottom : 

For  67°+  or  247°+, 
read  as  printed;  for 
157°+  or  337°+,  read 
co-function. 


29 

6 

2 

5.8 

1.2 

3 

8.7 

1.8 

4 

11.6 

2.4 

6 

14.5 

3.0 

6 

17.4 

3.6 

7 

20.3 

4.2 

8 

23.2 

4.8 

9 

26.1 

5.4 

L  Cos   d   L  Ctn   c  d  L  Tan 


L  Sin 


d   ' 


Prop.  Pts. 


67°  —  Losrarithnis  of  Trigonometric  Functions 


Ill] 


23°  —  Logarithms  of  Trigonometric  Functions 


69 


L  Sin 


L  Tan     c  d      L  Gtn 


L  Cos 


Prop.  Pts. 


0 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 

45 

46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


9.59188 
9.59  218 
9.59  247 
9.59  277 
9.59  307 
9.59  336 
9.59  366 
9.59  396 
9.59  425 
9.59455 
9.59  484 
9.59  514 
9.59  543 
9.59  573 
9.59  602 
9.59  632 
9.59  661 
9.59  690 
9.59  720 
9.59  749 
9.59  778 
9.59  808 
9.59  837 
9.59  866 
9.59  895 
9.59  924 
9.59  954 

9.59  983 

9.60  012 
9.60  041 
9.60  070 
9.60  099 
9.60  128 
9.60  157 
9.60  186 
9.60  215 
9.60  244 
9.60  273 
9.60  302 
9.60  331 
9.60  359 
9.60  388 
9.60  417 
9.60  446 
9.60  474 
9.60  503 
9.60  532 
9.60  561 
9.60  589 
9.60  618 
9.60  646 
9.60  675 
9.60  704 
9.60  732 
9.60  761 
9.60  789 
9.60  818 
9.60  84(5 
9.60  875 
9.60  903 
9.60  931 


9.62  785 
9.62  820 
9.62  855 
9.62  890 
9.62  926 
9.62  961 

9.62  996 
9.63031 

9.63  066 
9.63 101 
9.63  135 
9.63170 
9.63  205 
9.63  240 
9.63  275 
9.63  310 
9.63  345 
9.63  379 
9.63  414 
9.63  449 
9.63  484 
9.63  519 
9.63  553 
9.63  588 
9.63  623 
9.63  657 
9.63  692 
9.63  726 
9.63  761 
9.63  796 
9.63  830 
9.63  865 
9.63  899 
9.63  934 

9.63  968 

9.64  003 
9.64  037 
9.64  072 
9.64  106 
9.64  140 
9.64  175 
9.()4  209 
9.64  243 
9.64  278 
9.64  312 
9.64  346 
9.64  381 
9.64  415 
9.64  449 
9.64  483 
9.64  517 
9.64  552 
9.64  586 
9.64  620 
9.64  654 
9.64  688 
9.64  722 
9.64  756 
9.64  790 
9.61824 
9.64  858 


0.37  215 
0.37  180 
0.37  145 
0.37  110 
0.37  074 
0.37  039 
0.37  004 
0.36  969 
0.36  934 
0.36  899 
0.36  865 
0.36  830 
0.36  795 
0.36  760 
0.36  725 
0.36  im 
0.36  655 
0.36  621 
0.36  586 
0.36  551 
0.36  516 
0.36  481 
0.36  447 
0.36  412 
0.36  377 
0.36  343 
0.36  308 
0.36  274 
0.36  239 
0.36  204 
0.36  170 
0.36  135 
0.36  101 
0.36  066 
0.36  032 
0.35  997 
0.35  963 
0.35  928 
0.35  894 
0.35  860 
0.35  825 
0.35  791 
0.35  757 
0..35  722 
0.35  688 
0.35  654 
0.35  619 
0.35  585 
0.35  551 
0.35  517 
0.35  483 
0.35  448 
0.35  414 
0.35  380 
0.35  346 
0.35  312 
0.35  278 
0.35  244 
0.35  210 
0.35  176 
0.35  142 


9.96  403 
9.96  397 
9.96  392 
9.96  387 
9.96  381 
9.96  376 
9.96  370 
9.96  365 
9.96  360 
9.96  354 
9.96  349 
9.96  343 
9.96  338 
9.96  333 
9.96  327 
9.96  322 
9.96  316 
9.96  311 
9.96  305 
9.96  300 
9.96  294 
9.96  289 
9.96  284 
9.96  278 
9.96  273 
9.96  267 
9.96  262 
9.96  256 
9.96  251 
9.96  245 
9.96  240 
9.96  234 
9.96  229 
9.96  223 
9.96  218 
9.96  212 
9.96  207 
9.96  201 
9.96  196 
9.96  190 
9.96185 
9.96  179 
9.96  174 
9.96  168 
9.96  162 
9.96  157 
9.96  151 
9.96  146 
9.96  140 
9.96  135 
9.96  129 
9.96  123 
9.96  118 
9.96112 
9.96  107 
9.96101 
9.96  095 
9.96  090 
9.96  084 
9.96  079 
9.96  073 


36 

35 

2 

7.2 

7.0 

3 

10.8 

10.5 

4 

14.4 

14.0 

5 

18.0 

17.5 

6 

21.6 

21.0 

7 

25.2 

24.5 

8 

28.8 

28.0 

9 

32.4 

31.5 

30 

29 

2 

6.0 

5.8 

3 

9.0 

8.7 

4 

12.0 

11.6 

5 

15.0 

14.5 

6 

18.0 

17  4 

7 

21.0 

20.3 

8 

24.0 

23.2 

9 

27.0 

26.1 

34 

6.8 
10.2 
13.6 
17.0 
20.4 
23.8 
27.2 
30.6 


5.6 
8.4 
11.2 
140 
16.8 
19.6 
22.4 
25.2 


2 

6 

1.2 

3 

1.8 

4 

2.4 

5 

3.0 

6 

3.6 

7 

4.2 

8 

4.8 

9 

6.4 

1.0 
1.5 
2.0 
2.5 
3.0 
3.5 
4.0 
4.5 


From  the  top  : 

For  23°+  or  203°+, 
read  as  printed;  for 
113°+  or  293°+,  read 
co-function. 

From  the  bottom : 

For  66°+  or  246°+, 

read  as  printed ;  for 
156°+  or  336°+,  read 
co-function. 


L  Cos 


LGtn 


c  d   L  Tan 


L  Sin 


Prop.  Pts. 


66° — Logarithms  of  Trigonometric  Functions 


70         34°  — Logarithms  of  TrigonomeMc  Functions         [in 


LSin 


L  Tan     c  d      L  Gtn 


L  Cos 


Prop.  Pts. 


0 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
63 
54 
55 
56 
57 
58 
59 
60 


9.60  931 
9.60  960 

9.60  988 
9.61016 
9.61045 
9.61073 
9.61 101 
9.61 129 
9.61 158 
9.61 186 
9.61214 
9.61 242 
9.61 270 

9.61  298 
9.61  326 
9.61 354 
9.61  382 
9.61 411 
9.61 438 
9.61466 
9.61 494 
9.61 522 
9.61 550 
9.61 578 
9.61 606 
9.61 634 
9.61 662 
9.61 689 
9.61  717 
9.61 745 
9.61 773 
9.61  800 
9.61  828 
9.61  856 
9.61 883 

9.61  911 
9.61 939 
9.61 966 
9.61 994 

9.62  021 
9.62.049 
9.62  076 
9.62  104 
9.62  131 
9.62  159 
9.62 186 
9.62  214 
9.62  241 
9.62  268 
9.62  296 
9.62  323 
9.62  350 
9.62  377 
9.62  405 
9.62  432 
9.62  459 
9.62  486 
9.62  513 
9.62  541 
9.62  568 
9.62  595 


9.64  858 
9.64  892 
9.64  926 
9.64  960 

9.64  994 

9.65  028 
9.65  062 
9.65  096 
9.65  130 
9.65  164 
9.65 197 
9.65  231 
9.65  265 
9.65  299 
9.65  333 
9.65  366 
9.65  400 
9.65  434 
9.65  467 
9.65  501 
9.65  535 
9.65  568 
9.65  602 
9.65  636 
9.65  669 
9.65  703 
9.65  736 
9.65  770 
9.65  803 
9.65  837 
9.65  870 
9.65  904 
9.65  937 

9.65  971 

9.66  004 
9.66  038 
9.66  071 
9.66 104 
9.66 138 
9.66 171 
9.66204 
9.66  238 
9.66  271 
9.66  304 
9.66  337 
9.66  371 
9.66  404 
9.66  437 
9.66  470 
9.66503 
9.66  537 
9.66  570 
9.66  603 
9.66  636 
9.66  669 
9.66  702 
9.66  735 
9.66  768 
9.66  801 
9.66  834 
9.66  867 


0.35  142 
0.35  108 
0.35074 
0.35  040 
0.35  006 
0.34  972 
0.34  938 
0.34  904 
0.34  870 
0.34  836 
0.34  803 
0.34  769 
0.34  735 
0.34  701 
0.34  667 
0.34  634 
0.34  600 
0.34  566 
0.34  533 
0.34  499 
0.34  465 
0.34  432 
0.34  398 
0.34  364 
0.34  331 
0.34  297 
0.34  264 
0.34  230 
0.34  197 
0.34  163 
0.34130 
0.34  096 
0.34  063 
0.34  029 
0.33  996 
0.33  962 
0.33  929 
0.33  896 
0.33  862 
0.33  829 
0.33  796 
0.33  762 
0.33  729 
0.33  696 
0.33  663 
0.33629 
0.33  596 
0.33  563 
0.33  530 
0.33497 
0.33463 
0.33430 
0.33  397 
0.33  364 
0.33  331 
0.33  298 
0.33  265 
0.33  232 
0.33 199 
0.33 166 
0.33  133 


9.96  073 
9.96067 
9.96  062 
9.96  056 
9.96050 
9.96  045 
9.96  039 
9.96034 
9.96  028 
9.96  022 
9.96  017 
9.96  Oil 
9.96  005 
9.96  000 
9.95  994 
9.95  988 
9.95  982 
9.95  977 
9.95  971 
9.95  965 
9.95  960 
9.95  954 
9.95  948 
9.95  942 
9.95  937 
9.95  931 
9.95  925 
9.95  920 
9.95  914 
9.95  908 
9.95  902 
9.95  897 
9.95  891 
9.95  885 
9.95  879 
9.95  873 
9.95  868 
9.95  862 
9.95  856 
9.95  850 
9.95  844 
9.95  839 
9.95  833 
9.95  827 
9.95  821 
9.95  815 
9.95  810 
9.95  804 
9.95  798 
9.95  792 
9.95  786 
9.95  780 
9.95  775 
9.95  769 
9.95  763 
9.95  757 
9.95  751 
9.95  745 
9.95  739 
9.95  733 
9.95  728 


34 

33 

2 

6.8 

6.6 

3 

10.2 

9.9 

4 

13.6 

13.2 

5 

17.0 

16.5 

6 

20.4 

19.8 

7 

23.8 

23.1 

8 

27.2 

26.4 

9 

30.6 

29.7 

29 

5.8 
8.7 
11.6 
14.5 
17.4 
20.3 
23.2 
26.1 


28 

27 

2 

5.6 

5.4 

3 

8.4 

8.1 

4 

11.2 

10.8 

5 

14.0 

13.5 

6 

16.8 

16.2 

7 

19.6 

18.9 

8 

22.4 

21.6 

9 

25.2 

24.3 

6 

2 

1.2 

3 

1.8 

4 

2.4 

5 

3.0 

6 

3.6 

7 

4.2 

8 

4.8 

9 

5.4 

1.0 
1.5 
2.0 
2.5 
3.0 
3.5 
4.0 
4.5 


From  the  top : 

For  24°+  or  204°+, 
read  as  printed;  for 
114°+ or  294°+,  read 
co-function. 

From  the  bottom : 

For  65°+  or  245°+, 

read  as  printed;  for 
155°+  or  335°+,  read 
co-function. 


L  Cos 


LCtn 


cd 


LTan 


L  Sin 


d     / 


Prop.  Pts. 


6/i°  —  TiOerarifliiris  of  Tri  iron  om  ft  trie  Functions 


nq 


25°  —  Logarithms  of  Trigonometric  Functions 


71 


LSin 


L  Tan    led     L  Gtn 


L  Cos 


Prop.  Pts. 


10 

11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


9.62  595 
9.62  622 
9.62  649 
9.62  676 
9.62  703 
9.62  730 
9.62  757 
9.62  784 
9.62  811 
9.62  838 
9.62  865 
9.62  892 
9.62  918 
9.62  945 
9.62  972 

9.62  999 

9.63  026 
9.63  052 
9.63  079 
9.63  106 
9.63  133 
9.63  159 
9.63 186 
9.63  213 
9.63  239 
9.63  266 
9.63  292 
9.63  319 
9.63  345 
9.63  372 
9.63  398 
9.63  425 
9.63451 
9.63  478 
9.63  504 
9.63  531 
9.63  557 
9.63  583 
9.63  610 
9.63636 
9.63  662 
9.63689 
9.63  715 
9.63  741 
9.63  767 
9.63  794 
9.63  820 
9.63  846 
9.63  872 
9.63  898 
9.63  924 
9.63  950 

9.63  976 
9.64002 
9.64028 

9.64  054 
9.64080 
9.64 106 
9.64  132 
9.64  158 
9.64  184 


LGos 


9.66  867 
9.66  900 
9.66  933 
9.66  966 

9.66  999 

9.67  032 
9.67  065 
9.67  098 
9.67  131 
9.67  163 
9.67  196 
9.67  229 
9.67  262 
9.67  295 
9.67  327 
9.67  360 
9.67  393 
9.67  426 
9.67  458 
9.67  491 
9.67  524 
9.67  556 
9.67  589 
9.67  622 
9.67  654 
9.67  687 
9.67  719 
9.67  752 
9.67  785 
9.67  817 
9.67  850 
9.67  882 
9.67  915 
9.67  947 

9.67  980 

9.68  012 
9.68  044 
9.68  077 
9.68  109 
9.68  142 
9.68  174 
9.68  206 
9.68  239 
9.68  271 
9.68  303 
9.68  336 
9.68  368 
9.68  400 
9.68  432 
9.68  465 
9.68  497 
9.68  529 
9.68  561 
9.68  593 
9.68  626 
9.68  658 
9.68  690 
9.68  722 
9.68  754 
9.68  786 
9.68  818 


L  Ctn   c  d  L  Tan 


0.33  133 
0.33  100 
0.33067 
0.33  034 
0.33  001 
0.32  968 
0.32  935 
0.32  902 
0.32  869 
0.32  837 
0.32  804 
0.32  771 
0.32  738 
0.32  705 
0.32  673 
0.32  640 
0.32  607 
0.32  574 
0.32  542 
0.32  509 
0.32  476 
0.32  444 
0.32  411 
0.32  378 
0.32  346 
0.32  313 
0.32  281 
0.32  248 
0.32  215 
0.32  183 
0.32  150 
0.32  118 
0.32  085 
0.32  053 
0.32  020 
0.31  988 
0.31  956 
0.31  923 
0.31  891 
0.31  858 
0.31  826 
0.31  794 
0.31  761 
0.31  729 
0.31  697 
0.31  664 
0.31  632 
0.31  600 
0.31  568 
0.31  535 
0.31  503 
0.31  471 
0.31  439 
0.31  407 
0.31  374 
0.31  342 
0.31  310 
0.31  278 
0.31246 
0.31  214 
0.31 182 


9.95  728 
9.95  722 
9.95  716 
9.95  710 
9.95  704 
9.95  698 
9.95  692 
9.95  686 
9.95  680 
9.95  674 
9.95  668 
9.95  663 
9.95  657 
9.95  651 
9.95  645 
9.95  639 
9.95  633 
9.95  627 
9.95  621 
9.95  615 
9.95  609 
9.95  603 
9.95  597 
9.95  591 
9.95  585 
9.95  579 
9.95  573 
9.95  567 
9.95  561 
9.95  555 
9.95  549 
9.95  543 
9.95  537 
9.95  531 
9.95  525 
9.95  519 
9.95  513 
9.95  507 
9.95  500 
9.95  494 
9.95  488 
9.95  482 
9.95  476 
9.95  470 
9.95  464 
9.95  458 
9.95  452 
9.95  446 
9.95  440 
9.95  434 
9.95  427 
9.95  421 
9.95  415 
9.95  409 
9.95  403 
9.95  397 
9.95  391 
9.95  384 
9.95  378 
9.95  372 
9.95  366 


L  Sin 


33 

32 

2 

6.6 

6.4 

3 

9.9 

9.6 

4 

13.2 

12.8 

5 

16.5 

16.0 

6 

19.8 

19.2 

7 

23.1 

22.4 

8 

26.4 

25.6 

9 

29.7 

28.8 

27 

5.4 
8.1 
10.8 
13.5 
16.2 
18.9 
21.6 
24.3 


d  ' 


26 

2 

5.2 

3 

7.8 

4 

10.4 

5 

13.0 

6 

15.6 

7 

18.2 

8 

20.8 

9 

23.4 

6 

2 

1.2 

3 

1.8 

4 

2.4 

5 

3.0 

6 

3.6 

7 

4.2 

8 

4.8 

9 

5.4 

7 

1.4 
2.1 
2.8 
3.5 
4.2 
4.9 
5.6 
6.3 


1.0 
1.5 
2.0 
2.5 
3.0 
3.5 
4.0 
4.5 


From  the  top : 

For  25°+  or  205^+, 

read  as  printed;  for 
115°+ or  295°+,  read 
co-functiou. 

From  the  bottom: 

For  64°+  or  244°+, 
read  as  printed  ;  for 
154°+ or  334°+,  read 
co-function. 


Prop.  Pts. 


64°— Logarithms  of  Trigonometric  Functions 


72  26°— Logarithms  of  Trigonometric  Functions         [ii, 


L  Sin 


LTan 


c  d      L  Gtn 


LCos 


Prop.  Pts. 


7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 

20 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


9.64  184 
9.64  210 
9.64  236 
9.64  262 
9.64  288 
9.64  313 
9.64  339 
9.64  3(i5 
9.64  391 
9  64  417 
9.64  442 
9.61468 
964  494 
9.64  519 
9.64  545 
9.64  571 
9.64  596 
9.64  622 
9.64  647 
9.64  673 
9.64  698 
9.64  724 
9.64  749 
9.64  775 
9.64  800 
9.64  826 
9.64  851 
9.64  877 
9.64  902 
9.64  927 
9.64  953 

9.64  978 

9.65  003 
9.65  029 
9.65  054 
9.65  079 
9.65  104 
9.65  130 
9.65  155 
9.65  180 
9.65  205 
9.65  230 
9.65  255 
9.65  281 
9.65  306 
9.65  331 
9.65  356 
9.65  381 
9.65  406 
9.65  431 
9.65  456 
9.65  481 
9.65  506 
9.65  531 
9.65  556 
9.65  580 
9.65  605 
9.65  630 
9.65  655 
9.65  680 
9.65  705 


26 

26 

26 

26 

25 

26 

26 

26 

26 

25 

26 

26 

25 

26 

26 

25 

26 

25 

26 

25 

26 

25 

26 

25 

26 

25 

26 

25 

25 

26 

25 

25 

26 

25 

25 

25 

26 

25 

25 

25 

25 

25 

26 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

24 

25 

25 

25 

25 

25 


9.68  818 
9.()8  850 
9.68  882 
9.68  914 
9.68  94() 

9.68  978 
9.69010 

9.69  042 
9.69  074 
9.69  106 
9.69 138 
9.69  170 
9.69  202 
9.69  234 
9.69  266 
9.69298 
9.69  329 
9.69  361 
9.69  393 
9.69425 
9.69457 
9.69488 
9.69  520 
9.69  552 
9.69  584 
9.69  615 
9.69  647 
9.69  679 
9.69  710 
9.69  742 
9.69  774 
9.69  805 
9.69  837 
9.69  868 
9.69  900 
9.69  932 
9.69  963 

9.69  995 

9.70  026 
9.70  058 
9.70  089 
9.70  121 
9.70  152 
9.70  184 
9.70  215 
9.70  247 
9.70  278 
9.70  309 
9.70  341 
9.70  372 
9.70  404 
9.70  435 
9.70  466 
9.70  498 
9.70  529 
9J0  560 
9.70  592 
9  JO  623 
9.70  654 
9.70  685 
9.70  717 


32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
31 
32 
32 
32 
32 
31 
32 
32 
32 
31 
32 

32 

31 

32 

32 

31 

32 

31 

32 

32 

31 

32 

31 

32 

31 

32 

31 

32 

31 

32 

31 

31 

32 

31 

32 

31 

31 

32 

31 

31 

32 

31 

31 

31 

32 


0.31 182 
0.31 150 
0.31 118 
0.31  086 
0.31 054 
0.31  022 
0.30  990 
0.30  958 
0.30  926 
0.30  894 
0.30  862 
0.30  830 
0.30  798 
0.30  766 
0.30  734 
0.30  702 
0.30  671 
0.30  639 
0.30  607 
0.30  575 
0.30  543 
0.30  512 
0.30  480 
0.30  448 
0.30  416 
0.30  385 
0.30  353 
0.30  321 
0.30  290 
0.30  258 
0.30  226 
0.30  195 
0.30  163 
0.30  132 
0.30  100 
0.30068 
0.30  037 
0.30  005 
0.29  974 
0.29  942 
0.29  911 
0.29  879 
0.29  848 
0.29  816 
0.29  785 
0.29  753 
0.29  722 
0.29  691 
0.29  659 
0.29  628 
0.29  596 
0.29  565 
0.29  534 
0.29  502 
0.29  471 
0.29  440 
0,29  408 
0.29  377 
0.29  346 
0.29  315 
0.29  283 


9.95  366 
9.95  360 
9.95  354 
9.9.)  348 
9.95  311 
9.95  335 
9.95  329 
9.95  323 
9.95  317 
9.95  310 
9.95  304 
9.95  298 
9.95  292 
9.95  286 
9.95  279 
9.95  273 
9.95  267 
9.95  261 
9.95  254 
9.95  248 
9.95  242 
9.95  236 
9.95  229 
9.95  223 
9.95  217 
9.95  211 
9.95  204 
9.95  198 
9.95  192 
9.95  185 
9.95  179 
9.95  173 
9.95  167 
9.95  160 
9.95  154 
9.95  148 
9.95  141 
9.95  135 
9.95  129 
9.95  122 
9.95116 
9.95  110 
9.95  103 
9.95  097 
9.95  090 
9.95  084 
9.95  078 
9.95  071 
9.95  065 
9.95  059 
9.95  052 
9.95  046 
9.95  039 
9.95  033 
9.95027 
9.95  020 
9.95  014 
9.95  007 
9.95  001 
9.94  995 
9.94  988 


60 

59 
58 
57 
56 
55 
54 
53 
62 
51 
50 
49 
48 
47 
46 

45 
44 
43 
42 
41 

40 
39 
38 
37 
36 

35 
34 
33 
32 
31 

30 

29 
28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 
9 


32 

31 

2 

64 

6.2 

3 

9.6 

9.3 

4 

12.8 

12.4 

5 

16.0 

15.5 

6 

19.2 

18.6 

7 

22.4 

21,7 

8 

25.6 

24.8 

9 

28.8 

27.9 

26 

5.2 

7.8 
10.4 
13.0 
15.6 
18.2 
20.8 
23.4 


25 

2 

5.0 

3 

7.5 

4 

10.0 

5 

12.5 

6 

15.0 

7 

17.5 

8 

20.0 

9 

22.5 

1.4 
2.1 
2.8 
3.5 
4.2 
4.9 
5.6 
6.3 


24 

4.8 
7.2 
9.6 
12.0 
14.4 
16.8 
19.2 
21.6 


6 

1.2 

1.8  ' 

2.4 

3.0 

3.6 

4.2 

4.8 

5.4 


From  the  top : 

For  26°+ or  206°+, 

read,  as  printed  ;  for 
116°+ or  296°+,  read 
co-function. 

From  the  bottom : 

For  63°+  or  243°+, 

read  as  printed;  for 
153°+  or  333°+,  read 
co-function. 


L  Cos 


L  Ctn      c  d     L  Tan 


L  Sin 


Prop.  Pts. 


63°  —  Logarithms  of  Trigonometric  Functions 


Ill]         27°  —  Logarithms  of  Trigonometric  Functions 


73 


LSin 


L  Tan 


cd 


LCtn 


L  Cos 


Prop.  Pts. 


0 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
.57 
58 
59 
60 


9.()5  705 
9.65  729 
9.65  754 
9.65  779 
9.65  804 
9.65  828 
9.65  853 
9.65  878 
9.65  902 
9.65  927 
9.65  952 

9.65  976 
9.66001 

9.66  025 
9.66  050 
9.66  075 
9.66  099 
9.r>6  124 
9.66  148 
9.66  173 
9.66  197 
9.66  221 
9.66  246 
9.66  270 
9.66  295 
9.66  319 
9.66  343 
9.66  368 
9.66  392 
9.66  416 
9.66  441 
9.66  465 
9.66  489 
9.66  513 
9.66  537 
9.66  562 
9.66  586 
9.66  610 
9.66  634 
9.66  658 
9.66  682 
9.66  706 
9.66  731 
9.66  755 
9.66  779 
9.66  803 
9.66  827 
9.66  851 
9.66  875 
9.66  899 

m  922 
66  946 
66  970 

66  994 
.67  018 

67  042 
67  066 
67  090 
.67  113 

67  137 
67  161 


9.70  717 
9.70  748 
9.70  779 
9.70  810 
9.70  841 
9.70  873 
9.70  904 
9.70  935 
9.70  966 

9.70  997 
9.71 028 
9.71 059 
9.71 090 
9.71 121 
9.71 153 
9.71 184 

9.71  215 
9.71  246 
9.71  277 
9.71  308 
9.71  339 
9.71  370 
9.71  401 
9.71  431 
9.71  462 
9.71  493 
9.71  524 
9.71  555 
9.71  586 
9.71  617 
9.71  648 
9.71679 
9.71709 
9.71  740 
9.71  771 
9.71  802 
9.71  833 
9.71  863 
9.71  894 
9.71  925 
9.71  955 

9.71  986 

9.72  017 
9.72  048 
9.72  078 
9.72 109 
9.72  140 
9.72  170 
9.72  201 
9.72  231 
9.72  262 
9.72  293 
9.72  323 
9.72  354 
9.72  384 
9.72  415 
9.72  445 
9.72  476 
9.72  506 
9.72  537 
9.72  567 


0.29  283 
0.29  252 
0.29  221 
0.29  190 
0.29  159 
0.29 127 
0.29  096 
0.29065 
0.29034 
0.29003 
0.28  972 
0.28  941 
0.28  910 
0.28  879 
0.28  847 
0.28  816 
0.28  785 
0.28  754 
0.28  723 
0.28  692 
0.28  661 
0.28  630 
0.28  599 
0.28  569 
0.28  538 
0.28  507 
0.28  476 
0.28  445 
0.28  414 
0.28  383 
0.28  352 
0.28  321 
0.28  291 
0.28  260 
0.28  229 
0.28  198 
0.28  167 
0.28  137 
0.28  106 
0.28  075 
0.28  045 
0.28  014 
0.27  983 
0.27  952 
0.27  922 
0.27  891 
0.27  860 
0.27  830 
0.27  799 
0.27  769 
0.27  738 
0.27  707 
0.27  677 
0.27  646 
0.27  616 
0.27  585 
0.27  555 
0.27  524 
0.27  494 
0.27  463 
0.27  433 


9.94  988 
9.94  982 
9.94  975 
9.94  969 
9.94  962 
9.94  956 
9.94  949 
9.94  943 
9.94  936 
9.94  930 
9.94  923 
9.94  917 
9.94  911 
9.94  904 
9.94  898 
9.94  891 
9.94  885 
9.94  878 
9.94  871 
9.94  865 
9.94  858 
9.91  852 
9.94  845 
9.94  839 
9.94  832 
9.94  826 
9.94  819 
9.94  813 
9.94  806 
9.94  799 
9.94  793 
9.94  786 
9.94  780 
9.94  773 
9.94  767 
9.94  760 
9.94  753 
9.94  747 
9.94  740 
9.94  734 
9.94  727 
9.94  720 
9.94  714 
9.94  707 
9.94  700 
9.94  694 
9.94  687 
9.94  680 
9.94  674 
9.94  667 
9.94  660 
9.94  654 
9.94  647 
9.94  640 
9.94  634 
9.^)4  627 
9.94  620 
9.94  614 
9.94  607 
9.94600 
9.94  593 


32 

31 

2 

6.4 

6.2 

3 

9.6 

9.3 

4 

12.8 

12.4 

5 

16.0 

15.5 

6 

19.2 

18.6 

7 

22.4 

21.7 

8 

25.6 

24.8 

9 

28.8 

27.9 

25 

24 

2 

5.0 

4.8 

3 

7.5 

7.2 

4 

10.0 

9.6 

5 

12.5 

12.0 

6 

15.0 

14.4 

7 

17.5 

16.8 

8 

20.0 

19.2 

9 

22.5 

21.6 

30 

6.0 
9.0 
12.0 
15.0 
18.0 
21.0 
24.0 
27.0 


23 

4.6 
6.9 
9.2 
11.5 
13.8 
16.1 
18.4 
20.7 


7 

2 

1.4 

3 

2.1 

4 

2.8 

5 

3.5 

6 

4.2 

7 

4.9 

8 

5.6 

9 

6.3 

1.2 

1.8 
2.4 
3.0 
3.6 
4.2 
4.8 
6.4 


From  the  top : 

For  27°+  or  207°+, 
read  as  printed;  for 
117°+  or  297°+,  read 
co-function. 

From  the  bottom : 

For  62°+  or  242°+, 

read  as  printed;  for 
152°+  or  332°+,  read 
co-function. 


L  Cos 


LCtn 


cd 


L  Tan 


L  Sin 


Prop.  Pts. 


63°— Logarithms  of  Trigonometric  Functions 


74 


38  —  Logarithms  of  Trigonometric  Functions         pii 


'       LSin 


L  Tan     c  d      L  Ctn 


LGos 


Prop.  Pts. 


0 

1 
2 

3 
4 
5 

6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


9.67161 
9.67  185 
9.67  208 
9.67  232 
9.67  256 
9.67  280 
9.67  303 
9.67  327 
9.67  350 
9.67  374 
9.67  398 
9.67  421 
9.67  445 
9.67  468 
9.67  492 
9.67  515 
9.67  539 
9.67  562 
9.67  586 
9.67  609 
9.67  633 
9.67  656 
9.67  680 
9.67  703 
9.67  726 
9.67  750 
9.67  773 
9.67  796 
9.67  820 
9.67  843 
9.67  866 
9.67  890 
9.67  913 
9.67  936 
9.67  959 

9.67  982 

9.68  006 
9.68  029 
9.68052 
9.68  075 
9.68098 
9.68 121 
9.68  144 
9.68  167 
9.68  190 
9.68  213 
9.68  237 
9.68  260 
9.68  283 
9.68  305 
9.68  328 
9.68  351 
9.68  374 
9.68  397 
9.68  420 
9.68443 
9.68466 
9.68  489 
9.68  512 
9.68  534 
9.68  557 


9.72  567 
9.72  598 
9.72  628 
9.72659 
9.72689 
9.72  720 
9.72  750 
9.72  780 
9.72  811 
9.72  841 
9.72  872 
9.72  902 
9.72  932 
9.72  963 

9.72  993 
9.73023 

9.73  054 
9.73084 
9.73 114 
9.73  144 
9.73  175 
9.73  205 
9.73  235 
9.73  265 
9.73  295 
9.73  326 
9.73  356 
9.73  386 
9.73  416 
9.73  446 
9.73  476 
9.73  507 
9.73  537 
9.73  567 
9.73  597 
9.73  627 
9.73  657 
9.73  687 
9.73  717 
9.73  747 

9.73  777 
9.73  807 
9.73  837 
9.73  867 
9.73  897 
9.73  927 
9.73  957 

9.73  987 

9.74  017 
9.74  047 
9.74  077 
9.74  107 
9.74  137 
9.74  166 
9.74  196 
9.74  226 
9.74  256 
9.74  286 
9.74  316 
9.74  345 
9.74  375 


0.27  433 
0.27  402 
0.27  372 
0.27  341 
0.27  311 
0.27  280 
0.27  250 
0.27  220 
0.27  189 
0.27  159 
0.27  128 
0.27  098 
0.27  068 
0.27  037 
0.27  007 
0.26  977 
0.26  946 
0.26  916 
0.26  886 
0.26  856 
0.26  825 
0.26  795 
0.26  765 
0.26  735 
0.26  705 
0.26674 
0.26  644 
0.26  614 
0.26  584 
0.26  554 
0.26  524 
0.26  493 
0.26  463 
0.26433 
0.26  403 
0.26  373 
0.26  343 
0.26  313 
0.26  283 
0.26  253 
0.26  223 
0.26  193 
0.26  163 
0.26  133 
0.26 103 
0.26073 
0.26  043 
0.26  013 
0.25  983 
0.25  953 
0.25  923 
0.25  893 
0.25  863 
0.25  834 
0.25  804 
0.25  774 
0.25  744 
0.25  714 
0.25  684 
0.25  655 
0.25  625 


9.94  593 
9.94  587 
9.94  580 
9.94  573 
9.94  567 
9.94  560 
9.94  553 
9.94  546 
9.94  540 
9.94  533 
9.94  526 
9.94  519 
9.94  513 
9.94  506 
9.94499 
9.94  492 
9.94  485 
9.94  479 
9.94  472 
9.94465 
9.94  458 
9.94  451 
9.94  445 
9.94438 
9.94  431 
9.94  424 
9.94  417 
9.94410 
9.94  404 
9.94  397 
9.94  390 
9.94  383 
9.94  376 
9.94  369 
9.94  362 
9.94  355 
9.94  349 
9.94  342 
9.94  335 
9.94328 
9.94  321 
9.94  314 
9.94  307 
9.94  300 
9.94  293 
9.94  286 
9.94  279 
9.94  273 
9.94  266 
9.94  259 
9.94  252 
9.94245 
9.94  238 
9.94  231 
9.94  224 
9.94  217 
9.94210 
9.94  203 
9.94  196 
9.94189 
9.94 182 


31 

30 

2 

6.2 

6.0 

3 

9.3 

9.0 

4 

12.4 

12.0 

5 

15.5 

15.0 

6 

18.6 

18.0 

7 

21.7 

21.0 

8 

24.8 

24.0 

9 

27.9 

27.0 

24 

23 

4.8 

4.6 

7.2 

6.9 

9.6 

9.2 

12.0 

11.5 

14.4 

13.8 

16.8 

16.1 

19.2 

18.4 

21.6 

20.7 

29 

5.8 
8.7 
11.6 
14.5 
17.4 
20.3 
23.2 
26.1 


22 

4.4 
6.6 
8.8 
11.0 
13.2 
15.4 
17.6 
19.8 


7 

2 

1.4 

3 

2.1 

4 

2.8 

5 

3.5 

6 

4.2 

7 

4.9 

8 

5.6 

9 

6.3 

6 

1.2 
1.8 
2.4 

3.0 
3.6 
4.2 

4.8 
5.4 


From  the  top : 

For  28°+  or  208°+, 

read  as  printed;  for 
118°+ or  298°+, read 
co-function. 

From  the  bottom : 

For  61°+  or  241°+, 

read  as  printed;  for 
151°+  or  331°+, read 
co-function. 


LGos 


LCtn 


cd 


LTan 


L  Sin 


Prop.  Pts. 


61°  — Logarithms  of  Trigonometric  Functions 


Ill] 


29°  —  Logarithms  of  Trigonometric  Functions 


75 


L  Sin 


LTan 


c  d     L  Ctn 


L  Cos 


Prop.  Pts. 


10 

11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
3(] 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


9.68  557 
9.68  580 
9.68  603 
9.68  625 
9.68  648 
9.68  671 
9.68  694 
9.68  716 
9.68  739 
9.68  762 
9.68  784 
9.68  807 
9.68  829 
9.68  852 
9.68  875 
9.68  897 
9.68  920 
9.68  942 
9.68  965 

9.68  987 
9.69010 

9.69  032 
9.69  055 
9.69  077 
9.69  100 
9.69  122 
9.69  144 
9.69 167 
9.69  189 
9.69  212 
9.69  234 
9.69  256 
9.69  279 
9.69  301 
9.69  323 
9.69  345 
9.69  368 
9.69  390 
9.69  412 
9.69  434 
9.69456 
9.69  479 
9.69  501 
9.69  523 
9.69  545 
9.69  567 
9.69  589 
9.69  611 
9.69  633 
9.69  655 
9.69  677 
9.69  699 
9.69  721 
9.69  743 
9.69  765 
9.69  787 
9.69  809 
9.69  831 
9.69  853 
9.69  875 
9.69  897 


9.74  375 
9.74  405 
9  74  435 
9.74  465 
9.74  494 
9.74  524 
9.74  554 
9.74  583 
9.74  613 
9.74  643 
9.74  673 
9.74  702 
9.74  732 
9.74  762 
9.74  791 
9.74  821 
9.74  851 
9.74  880 
9.74  910 
9.74  939 
9.74  969 

9.74  998 

9.75  028 
9.75  058 
9.75  087 
9.75  117 
9.75  14^ 
9.75  176 
9.75  205 
9.75  235 
9.75  264 
9.75  294 
9.75  323 
9.75  353 
9.75  382 
9.75  411 
9.75  441 
9.75  470 
9.75  500 
9.75  529 
9.75  558 
9.75  588 
9.75  617 
9.75  647 
9.75  676 
9.75  705 
9.75  735 
9.75  764 
9.75  793 
9.75  822 
9.75  852 
9.75  881 
9.75  910 
9.75  939 
9.75  969 

9.75  998 

9.76  027 
9.76  056 
9.76  086 
9.76  115 
9.76144 


0.25  625 
0.25  595 
0.25  565 
0.25  535 
0.25  506 
0.25  476 
0.25  446 
0.25  417 
0.25  387 
0.25  357 
0.25  327 
0.25  298 
0.25  268 
0.25  2;58 
0.25  209 
0.25  179 
0.25  149 
0.25  120 
0.25  090 
0.25  061 
0.25  031 
0.25  002 
0.24  972 
0.24  912 
0.24  913 
0.24  883 
0.24  854 
0  24  824 
0.24  795 
0.24  765 
0.24  736 
0.24  706 
0.24  677 
0.24  647 
0.24  618 
0.24  589 
0.24  559 
0.24  530 
0.24  500 
0.24  471 
0.24  442 
0.24  412 
0.24  383 
0.24  353 
0.24  324 
0.24  295 
0.24  265 
0.24  236 
0.24  207 
0.24  178 
0.24  148 
0.24  119 
0.24  090 
0.24(61 
0.24  031 
0.24  002 
0.23  973 
0.23  944 
0.23  914 
0.23  885 
0.23  856 


9.94  182 
9.94  175 
9.94 168 
9.94  161 
9.94  154 
9.94  147 
9.94  140 
9.94133 
9.94 126 
9.94 119 
9.94 112 
9.94  105 
9.94  098 
9.94  090 
9.94  083 
9.94  076 
9.94  069 
9.94  062 
9.94  055 
9.94  048 
9.94041 
9.94  034 
9.94  027 
9.94  020 
9.94  012 
9.94  005 
9.93  998 
9.93  991 
9.93  984 
9.93  977 
9.93  970 
9.93  963 
9.93  955 
9.93  948 
9.93  941 
9.93  934 
9.93  927 
9.93  920 
9.93  912 
9.93  905 
9.93  898 
9.93  891 
9.93  884 
9.93  876 
9.93  869 
9.93  862 
9.93  855 
9.93  847 
9.93  840 
9.93  833 
9.93  826 
9.93  819 
9.93  811 
9.93  804 
9.93  797 
9  93  789 
9.93  782 
9.93  775 
9.93  768 
9.93  760 
9.93  753 


30 

29 

2 

6.0 

5.8 

3 

9.0 

8.7 

4 

12.0 

11.6 

5 

15.0 

14.5 

6 

18  0 

17.4 

7 

21.0 

20.3 

8 

24.0 

23.2 

9 

27.0 

26.1 

22 

8 

2 

4.4 

1.6 

3 

6.6 

2.4 

4 

8.8 

3.2 

5 

11.0 

4.0 

6 

13.2 

4.8 

7 

15.4 

5.6 

8 

17.6 

6.4 

9 

19.8 

7.2 

4.6 
6.9 
9.2 
11.5 
13.8 
16.1 
18.4 
20.7 


7 

1.4 
2.1 
2.8 
3.5 
4.2 
4.9 
5.6 
6.3 


From  the  top  : 

For  29°+  or  209°+, 

read  as  printed;  for 
119°+  or  299°+,  read 
co-function. 


From  the  bottom: 

For  60°+  or  240°+, 

read  as  printed  ;    for 
150°+or  330°+,  read 

co-function. 


L  Cos 


L  Ctn    I  c  d 


LTan 


L  Sin 


Prop.  Pts. 


60°— Logarithms  of  Trigonometric  Functions 


76 


30°  — Logarithms  of  Trigonometric  Functions         [in 


L  Sin 


L  Tan     c  d      L  Gtn 


L  Cos 


Prop.  PtB. 


0 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


9.69  897 
9.69  919 
9.69941 
9.69  963 

9.69  984 

9.70  006 
9.70028 
9.70050 
9.70  072 
9.70  093 
9.70  115 
9.70  137 
9.70  159 
9.70  180 
9.70  202 
9.70  224 
9.70  245 
9.70  267 
9.70  288 
9.70  310 
9.70  332 
9.70  353 
9.70  375 
9.70  396 
9.70  418 
9.70439 
9.70  461 
9.70  482 
9.70  504 
9.70  525 
9.70  547 
9.70  568 
9.70  590 
9.70  611 
9.70  633 
9.70  654 
9.70  675 
9.70  697 
9.70  718 
9.70  739 
9.70  761 
9.70  782 
9.70  803 
9.70  824 
9.70  846 
9.70  867 
9.70  888 
9.70  909 
9.70  931 
9.70  952 
9.70  973 
9.70  994 
9.71 015 
9.71 036 
9.71 058 
9.71079 
9.71 100 
9.71 121 
9.71 142 
9.71 163 
9.71 184 


9.76  144 
9.76  173 
9.76  202 
9.76  231 
9.76  261 
9.76  290 
9.76  319 
9.76  348 
9.76  377 
9.76  406 
9.76  435 
9.76  464 
9.76  493 
9.76  522 
9.76  551 
9.76  580 
9.76  609 
9.76  639 
9.76  668 
9.76  697 
9.76  725 
9.76  754 
9.76  783 
9.76  812 
9.76  841 
9.76  870 
9.76  89^) 
9.76  928 
9.76  957 

9.76  986 

9.77  015 
9.77  044 
9.77  073 
9.77  101 
9.77  130 
9.77  159 
9.77  188 
9.77  217 
9.77  246 
9.77  274 
9.77  303 
9.77  3.32 
9.77  361 
9.77  390 
9.77  418 
9.77  447 
9.77  476 
9.77  505 
9.77  533 
9.77  562 
9.77  591 
9.77  619 
9.77  648 
9.77  677 
9.77  706 
9.77  734 
9.77  763 
9.77  791 
9.77  820 
9.77  849 
9.77  877 


0.23  856 
0.23  827 
0.23  798 
0.23  769 
0.23  739 
0.23  710 
0.23  681 
0.23  652 
0.23  623 
0.23  594 
0.23  565 
9.23  536 
0.23  507 
0.23  478 
0.23  449 
0.23420 
0.23  391 
0.23  361 
0.23  332 
0.23  303 
0.23  275 
0.23  246 
0.23  217 
0.23  188 
0.23  159 
0.23  130 
0.23  101 
0.23  072 
0.23  043 
0.23  014 
0.22  985 
0.22  956 
0.22  927 
0.22  899 
0.22  870 
0.22  841 
0.22  812 
0.22  783 
0.22  754 
0.22  726 
0.22  697 
0.22  668 
0.22  639 
0.22  610 
0.22  582 
0.22  553 
0.22  524 
0.22  495 
0.22  467 
0.22  438 
0.22  409 
0.22  381 
0.22  352 
0.22  323 
0.22  294 
0.22  266 
0.22  237 
0.22  209 
0.22  180 
0.22  151 
0.22  123 


9.93  753 
9.93  746 
9.93  738 
9.93  731 
9.93  724 
9.93  717 
9.93  709 
9.93  702 
9.93  695 
9.93  687 
9.93  680 
9.93  673 
9.93  665 
9.93  658 
9.93  650 
9.93  643 
9.93  636 
9.93  628 
9.93  621 
9.93  614 
9.93  606 
9.93  599 
9.93  591 
9.93  584 
9.93  577 
9.93  569 
9.93  562 
9.93  554 
9.93  547 
9.93  539 
9.93  532 
9.93  525 
9.93  517 
9.93  510 
9.93  502 
9.93  495 
9.93  487 
9.93480 
9.93  472 
9.93  465 
9.93  457 
9.93  450 
9.93  442 
9.93  435 
9.93  427 
9.93  420 
9.93  412 
9.93  405 
9.93  397 
9.93  390 
9.93  382 
9.93  375 
9.93  367 
9.93  360 
9.93  352 
9.93  344 
9.93  337 
9.93  329 
9.93  322 
9.93  314 
9.93  307 


30 

29 

2 

6.0 

5.8 

3 

9.0 

8.7 

4 

12.0 

11.6 

5 

15.0 

14.5 

6 

18.0 

17.4 

7 

21.0 

20.3 

8 

24.0 

23.2 

9 

27.0 

26.1 

28 

6.6 
8.4 
11.2 
14.0 
16.8 
19.6 
22.4 
25.2 


22 

2 

4.4 

3 

6.6 

4 

8.8 

5 

11.0 

6 

13.2 

7 

15.4 

8 

17.6 

9 

19.8 

8 

2 

1.6 

3 

2.4 

4 

3.2 

5 

4.0 

6 

4.8 

7 

5.6 

8 

6.4 

9 

7.2 

21 

4.2 
6.3 
8.4 
10.5 
12.6 
14.7 
16.8 
18.9 


7 

1.4 
2.1 
2.8 
3.5 
4.2 
4.9 
5.6 
6.3 


From  the  top  : 

For  30°+  or  210°+, 

read  as  printed  ;  for 
120°+  or  300°+,  read 
co-function. 

From  the  bottom : 

For  59°+  or  239°+, 

read  as  printed;  for 
149°+  or  329°+,  read 
co-function. 


L  Cos 


L  Ctn   c  d 


L  Tan 


L  Sin   d  I 


Prop.  Pts. 


59°  —  Losraritlims  of  Trigonometric  Functions 


Ill] 


31°  —  Logarithms  of  Trigonometric  Functions  77 


L  Sin 


L  Tan  c  d   L  Ctn 


L  Cos 


Prop.  Pts. 


0 

1 
2 
3 
4 

5 

6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
.57 
58 
59 
60 


9.71 184 
9.71 205 
9.71  226 
9.71  247 
9.71 268 
9.71  289 
9.71  310 
9.71  331 
9.71  352 
9.71  373 
9.71  393 
9.71 414 
9.71 435 
9.71 456 
9.71 477 
9.71498 
9.71 519 
9.71 539 
9.71  560 
9.71 581 
9.71  602 
9.71  622 
9.71  643 
9.71  664 
9.71 685 
9.71  705 
9.71  726 
9.71  747 
9.71  767 
9.71  788 
9.71  809 
9.71  829 
9.71  850 
9.71  870 
9.71  891 
9.71  911 
9.71  932 
9.71  952 

9.71  973 
9.71 994 

9.72  014 
9.72  03i 
9.72  055 
9.72  075 
9.72  096 
9.72  116 
9.72  137 
9.72  157 
9.72  177 
9.72  198 
9.72  218 
9.72  238 
9.72  259 
9.72  279 
9.72  299 
9.72  320 
9.72  340 
9.72  360 
9.72  381 
9.72  401 
9.72  421 


9.77  877 
9.77  906 
9.77  935 
9.77  9(53 

9.77  992 

9.78  020 
9.78  049 
9.78  077 
9.78  106 
9.78  135 
9.78  163 
9.78  192 
9.78  220 
9.78  249 
9.78  277 
9.78  306 
9.78  334 
9.78  363 
9.78  391 
9.78  419 
9.78  448 
9.78  476 
9.78  505 
9.78533 
9.78  uG2 
9.78  590 
9.78  618 
9.78  647 
9.78  675 
9.78  704 
9.78  732 
9.78  760 
9.78  789 
9.78  817 
9.78  845 
9.78  874 
9.78  902 
9.78  930 
9.78  959 

9.78  987 
9.79015 
9.79043 

9.79  072 
9.79  100 
9.79  128 
9.79  156 
9.79  185 
9.79  213 
9.79  241 
9.79  269 
9.79  2i)7 
9.79  326 
9.79  354 
9.79  382 
9.79  410 
9.79  438 
9.79  466 
9.79  495 
9.79  523 
9.79551 
9.79  579 


0.22  123 
0.22  094 
0.22  065 
0.22  037 
0.22  008 
0.21 980 
0.21951 
0.21  923 
0.21  894 
0.21  865 
0.21  837 
0.21  808 
0.21  780 
0.21 751 
0.21 723 
0.21694 
0.21 666 
0.21  637 
0.21  609 
0.21 581 
0.21  552 
0.21  524 
0.21  495 
0.21 467 
0.21 438 
0.21  410 
0.21 382 
0.21  353 
0.21  325 
0.21  296 
0.21  268 
0.21  240 
0.21  311 
0.21  383 
0.21 155 
0.21 126 
0.21  098 
0.21 070 
0.21 041 
0.21 013 
0.20  985 
0.20  957 
0  20  928 
0.20  900 
0.20  872 
0.20  844 
0.20  815 
0.20  787 
0.20  759 
0.20  731 
0.20  703 
0.20  674 
0.20  646 
0.20618 
0.20590 
0.20  562 
0.20  534 
0.20505 
0.20  477 
0.20  449 
0.20  421 


9.93  307 
9.93  299 
9.93  291 
9.93  284 
9.93  276 
9.93  269 
9.93  261 
9.93  253 
9.93  246 
9.93  238 
9.93230 
9.93  223 
9.93  215 
9.93  207 
9.93  200 
9.93 192 
9.93 184 
9.93 177 
9.93 169 
9.93 161 
9.93154 
9.93 146 
9.93 138 
9.93 131 
9.93  123 
9.93 115 
9.93 108 
9.93 100 
9.93  092 
9.93084 
9.93077 
9.93  069 
9.93  061 
9.93  053 
9.93  046 
9.93  038 
9.93030 
9.93  022 
9.93  014 
9.93  007 
9.92  999 
9.92  991 
9.92  983 
9.92  976 
9.92  968 
9.92  960 
Q.92  952 
9,92  944 
9.92  936 
9.92  929 
9.92  921 
9.92  913 
9.92  905 
9.92  897 
9.92  889 
9.92  881 
9.92  874 
9.92  866 
9.92  858 
9.92  850 
9.92  842 


60 

59 
58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 
10 
9 
8 
7 
6 
5 
4 
3 
2 
1 
0 


29 

28 

2 

5.8 

5.6 

3 

8.7 

8.4 

4 

11.6 

11.2 

5 

14.5 

14.0 

6 

17.4 

16.8 

7 

20.3 

19.6 

8 

23.2 

22.4 

9 

26.1 

25.2 

20 

8 

4.0 

1.6 

6.0 

2.4 

8.0 

3.2 

10.0 

4.0 

12.0 

4.8 

14.0 

5.6 

16.0 

6.4 

18.0 

7.2 

21 

4.2 
6.3 
8.4 
10.5 
1^.6 
14.7 
16.8 
18.9 


7 

1.4 
2.1 
2.8 
3.5 
4.2 
4.9 
5.6 
6.3 


From  the  top : 

For  31°+ or  211°+, 

read  as  printed;  for 
121°+  or  301°+,  read 
co-f  unctioD . 

From  the  bottom : 

For  58°+  or  238°+, 

read  as  printed;  for 
148°+  or  328°+, read 
co-function. 


LGos 


L  Ctn      c  d 


L  Tan 


L  Sin      d      ' 


Prop.  Pts. 


58°— Logarithms  of  Trigonometric  Functions 


78 


32°  —  Logarithms  of  Trigonometric  Functions         [in 


L  Sin 


L  Tan     c  d      L  Ctn 


LCos 


Prop.  Pts. 


10 

11 
12 
13 
14 
16 
16 
17 
IH 
19 
20 
21 
22 
23 
24 
25 
2() 
27 
28 
29 
30 
31 
32 
33 
34 
35 
3i) 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


9.72  421 
9.72  441 
9.72  4()1 
9.72  482 
9.72  502 
9.72  522 
9.72  542 
9.72  562 
9.72  582 
9.72  602 
9.72  622 
9.72  ()43 
9.72  663 
9.72  683 
9.72  703 
9.72  723 
9.72  743 
9.72  763 
9.72  783 
9.72  803 
9.72  823 
9.72  843 
9.72  863 
9  72  883 
9.72  902 
9.72  922 
9  72  942 
9.72  962 

9.72  982 

9.73  002 
9  73  022 
9.73  041 
9.73  061 
9.73  081 
9.73  101 
9.73  121 
9.73  140 
9.73  160 
9.73  180 
9.73  200 
9.73  219 
9.73  239 
9.73  259 
9.73  278 
9.73  298 
9.73  318 
9.73  337 
9.73  357 
9.73  377 
9.73  396 
9.73  416 
9.73435 
9.73455 
9.73  474 
9.73  494 
9.73513 
9.73  533 
9.73  552 
9.73  572 
9.73  591 
9.73  611 


9.79 
9.79 
9.79 
9  79 
9.79 
9.79 
9.79 
9.79 
9.79 
9.79 
9.79 
9.79 
9.79 
9.79 
9.79 
9.80 
9.80 
9.80 
9.8") 
9.80 
9.80 
9.80 
9.80 
9.80 
9.80 
9.80 
9.80 
9.80 
9.80 
9.80 
9.80 
9.80 
9.80 
9.80 
9.80 
9.80 
9.80 
9.80 
9.80 
9.80 
9.80 
9.80 
9.80 
9.80 
9.80 
9.80 
9.80 
9.80 
9.80 
9.80 
9.80 
9.81 
9.81 
9.81 
9.81 
9.81 
9.81 
9.81 
9.81 
9.81 
9.81 


579 
607 
(>35 
663 
691 
719 
747 
776 
804 
832 
860 
888 
916 
944 
972 
000 
028 
056 
084 
112 
140 
168 
195 
223 
251 
279 
307 
335 
3(53 
391 
419 
447 
474 
502 
530 
558 
586 
614 
642 
669 
697 
725 
753 
781 
808 
836 
864 
892 
919 
947 
975 
003 
030 
058 
086 
113 
141 
169 
196 
224 
252 


0.20  421 
0.20  393 
0.20  365 
0.20  337 
0.20  309 
0.20  281 
0.20  253 
0.20  224 
0.20  196 
0.20  168 
0.20  140 
0.20112 
0.20084 
0.20  056 
0.20  028 
0.20  000 
0.19  972 
0.19  944 
0.19  916 
0.19  888 
0.19  860 
0.19  832 
0.19  805 
0.19  777 
0.19  749 
0.19  721 
0.19  693 
0.19  665 
0.19  637 
0.19  609 
0.19  581 
0.19  553 
0.19  52() 
0.19  498 
0.19  470 
0.19  442 
0.19  414 
0.19  386 
0.19  358 
0.19  331 
0.19  303 
0.19  275 
0.19  247 
0.19  219 
0.19 192 
0.19164 
0.19136 
0.19  108 
0.19081 
0.19  053 
0.19  025 
0.18  997 
0.18  970 
0.18  942 
0.18  914 
0.18  887 
0.18  859 
0.18  831 
0.18  804 
0.18  776 
0.18  748 


9.92  842 
9.92  834 
9.92  826 
9.92  818 
9.92  810 
9.92  803 
9.92  795 
9.92  787 
9.92  779 
9.92  771 
9.92  763 
9.92  755 
9.92  747 
9.92  739 
9.92  731 
9.92  723 
9.92  715 
9.92  707 
9.92  699 
9.92  691 
9.92  683 
9.92  675 
9.92  667 
9.92  659 
9.92  651 
9.92  643 
9.92  635 
9.92  627 
9.92  619 
9.92  611 
9.92  603 
9.92  595 
9.92  587 
9.92  579 
9.92  571 
9.92  563 
9.92  555 
9.92  546 
9.92  538 
9.92  530 
9.92  522 
9.92  514 
9.92  506 
9.92  498 
9.92  490 
9.92  482 
9.92  473 
9.92  465 
9.92  457 
9.92  449 
9.92  441 
9.92  433 
9.92  425 
9.92  416 
9.92  408 
9.92  400 
9.92  392 
9.92  384 
9.92  376 
9.92  367 
9.92  359 


29 

28 

2 

5.8 

5.6 

3 

8.7 

8.4 

4 

11.6 

11.2 

5 

14.5 

14.0 

6 

17.4 

16.8 

7 

20.3 

19.6 

8 

23.2 

22.4 

9 

26.1 

25.2 

21 

20 

2 

4.2 

4.0 

3 

6.3 

6.0 

4 

8.4 

8.0 

5 

10.5 

10.0 

6 

12.6 

12.0 

7 

14.7 

14.0 

8 

16.8 

16.0 

9 

18.9 

18.0 

27 

5.4 

8.1 
10.8 
13.5 
16.2 
18.9 
21.6 
24.3 


19 

3.8 

5.7 

7.6 

9.5 

11.4 

13.3 

15.2 

17.1 


1.4 
2.1 
2.8 
3.5 
4.2 
4.9 
5.6 
6.3 


Frotn  the  top : 

For  32°+  or  212°+, 
read  as  printed;  for 
122°+  or  302°+,  read 
co-function. 

Froyn  the  "bottom : 

For  57°+  or  237°+, 
read  as  printed;  for 
147°+  or  327°+,  read 
co-function. 


9 

8 

2 

1.8 

1.6 

3 

2.7 

2.4 

4 

3.6 

3.2 

5 

4.5 

4.0 

6 

5.4 

4.8 

7 

6.3 

5.6 

8 

7.2 

6.4 

9 

8.1 

7.2 

LCos 


L  Ctn   c  d 


L  Tan 


L  Sin  Id 


Prop.  Pts. 


57°  — Logarithms  of  Trigonometric  Functions 


33°  — Logarithms  of  Trigonometric  Functions 


79 


L  Sin 


L  Tan ,    c  d     L  Ctn 


L  Cos 


Prop.  Pts. 


0 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
1() 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
3(3 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
'57 
58 
59 
60 


9.73  611 
9.73630 
9.73650 
9.73  669 
9.73  689 
9.73  708 
9.73  727 
9.73  747 
9.73  766 
9.73  785 
9.73  805 
9.73  824 
9.73  843 
9.73  863 
9.73  882 
9.73  901 
9.73  921 
9.73  940 
9.73  959 
9.73  978 

9.73  997 

9.74  017 
9.74036 
9.74  055 
9.74074 
9.74093 
9.74  113 
9.74 132 
9.74  151 
9.74 170 
9.74  189 
9.74  208 
9.74  227 
9.74246 
9.74265 
9.74  284 
9.74  303 
9.74  322 
9.74  341 
9.74  360 
9.74  379 
9.74  398 
9.74  417 
9.74  436 
9.74455 
9.74  474 
9.74  493 
9.74  512 
9.74  531 
9.74  549 
9.74568 
9.74  587 
9.74  606 
9.74  625 
9.74  644 
9.74  662 
9.74  681 
9.74  700 
9.74  719 
9.74  737 
9.74  756 


9.81 252 
9.81  279 
9.81  307 
9.81  335 
9.81  362 
9.81  390 
9.81  418 
9.81 445 
9.81  473 
9.81 500 
9.81528 
9.81 556 
9.81  583 
9.81611 
9.81 638 
9.81 666 
9.81 693 
9.81  721 
9.81 748 
9.81776 
9.81 803 
9.81  831 
9.81 858 
9.81  886 
9.81 913 
9.81  941 
9.81968 

9.81  996 

9.82  023 
9.82  051 
9.82  078 
9.82  106 
9.82  133 
9.82  161 
9.82  188 
9.82  215 
9.82  243 
9.82  270 
9.82  298 
9.82  325 
9.82  352 
9.82  380 
9.82  407 
9.82  435 
9.82  462 
9.82  489 
9.82  517 
9.82  544 
9.82  571 
9.82  599 
9.82  626 
9.82  653 
9.82  681 
9.82  708 
9.82735 
9.82  762 
9.82  790 
9.82  817 
9.82  844 
9.82  871 
9.82  899 


0.18  748 
0.18  721 
0.18  693 
0.18  665 
0.18  638 
0.18  610 
0.18  582 
0.18555 
0.18  527 
0.18  500 
0.18  472 
0.18  444 
0.18417 
0.18  389 
0.18  362 
0.18  334 
0.18  307 
0.18  279 
0.18  252 
0.18  224 
0.18  197 
0.18  169 
0.18  142 
0.18  114 
0.18  087 
0.18  059 
0.18  032 
0.18  004 
0.17  977 
0.17  949 
0.17  922 
0.17  894 
0.17  867 
0.17  839 
0.17  812 
0.17  785 
0.17  757 
0.17  730 
0.17  702 
0.17  675 
0.17648 
0.17  620 
0.17  593 
0.17  565 
0.17538 
0.17511 
0.17483 
0.17  456 
0.17  429 
0.17  401 
0.17  374 
0.17  347 
0.17  319 
0.17  292 
0.17  265 
0.17  238 
0.17  210 
0.17  183 
0.17  156 
0.17  129 
0.17  101 


9.92  359 
9.92  351 
9.92  343 
9.92  335 
9.92  326 
9.92  318 
9.92  310 
9.92  302 
9.92  293 
9.92  285 
9.92  277 
9.92  269 
9.92  260 
9.92  252 
9.92  244 
9.92  235 
9.92  227 
9.92  219 
9.92  211 
9.92  202 
9.92 194 
9.92  186 
9.92  177 
9.i;2  169 
9.92  161 
9.92  152 
9.92  144 
9.92  136 
9.92  127 
9.92  119 
9.92  111 
9.92  102 
9.92  094 
9.92  086 
9.92  077 
9.92  069 
9.92  060 
9.92  052 
9.92  044 
9.92  035 
9.92  027 
9.92  018 
9.92  010 
9.92002 
9.91 993 
9.91 985 
9.91  976 
9.91  968 
9.91 959 
9.91 951 
9.91942 
9.91 934 
9.91  925 
9.91 917 
9.91908 
9.91900 
9.91  891 
9.91 883 
9.91  874 
9.91 866 
9.91 857 


28 

27 

2 

5.6 

5.4 

3 

8.4 

8.1 

4 

11.2 

10.8 

5 

14.0 

13.5 

6 

16  8 

16.2 

7 

19.6 

18.9 

8 

22.4 

21.6 

9 

25.2 

24.3 

20 

40 
6.0 
8.0 
10.0 
12.0 
14.0 
16.0 
18.0 


19 

2 

3.8 

3 

5.7 

4 

7.6 

5 

9.5 

6 

11.4 

7 

13.3 

8 

15.2 

9 

17.1 

9 

2 

1.8 

3 

2.7 

4 

3.6 

5 

4.5 

6 

5.4 

7 

6.3 

8 

7.2 

9 

8.1 

18 

3.6 

5.4 

7.2 

9.0 

10.8 

12.6 

14.4 

16.2 


1.6 
2.4 
3.2 
4.0 
4.8 
5.6 
6.4 
7.2 


From  the  top : 

For  33°+  or  213°+, 

read  as  printed;  for 
123°+  or  303°+,  read 
co-function. 

From  the  bottom: 

For  66°+  or  236°+, 

read  as  printed;  for 
146°+  or  326°+,  read 
co-function. 


L  Cos 


LCtn 


c  d     L  Tan        L  Sin    |  d     ' 


Prop.  Pta. 


56°— Logarithms  of  Trigonometric  Functions 


80 


34°  — Logarithms  of  Trigonometric  Functions         [in 


'       LSin 


LTan 


c  d      L  Gtn 


L  Cos 


Prop.  Pts. 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

2$ 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 


9.74  756 
9.74  775 
9.74  794 
9.74  812 
9.74  831 
9.74  850 
9.74  868 
9.74  887 
9.74  906 
9.74  924 
9.74  943 
9.74  961 
9.74  980 

9.74  999 

9.75  017 
9.75  036 
9.75  054 
9.75  073 
9.75  091 
9.75 110 
9.75 128 
9.75 147 
9.75  165 
9.75 184 
9.75  202 
9.75  221 
9.75  239 
9.75  258 
9.75  276 
9.75  294 
9.75  313 
9.75  331 
9.75  350 
9.75  368 
9.75  386 
9.75  405 
9.75  423 
9.75  441 
9.75  459 
9.75  478 
9.75  496 
9.75  514 
9.75  533 
9.75  551 
9.75  569 

9.75  587 
9.75  605 
9.75  624 
9.75  642 
9.75  660 
9.75  678 
9.75  696 
9.75  714 
9.75  733 
9.75  751 
9.75  769 
9.75  787 
9.75  805 
9.75  823 
9.75  841 
9.75  859 


9.82  899 
9.82  926 
9.82  953 

9.82  980 
9.83008 

9.83  035 
9.83  062 
9.83  089 
9.83  117 
9.83  144 
9.83  171 
9.83  198 
9.83  225 
9.83  252 
9.83  280 
9.83  307 
9.83  334 
9.83  361 
9.83  388 
9.83  415 
9.83442 
9.83  470 
9.83  497 
9.83  524 
9.83  551 
9.83  578 
9.83  605 
9.83  632 
9.83  659 
9.83  686 
9.83  713 
9.83  740 
9.83  768 
9.83  795 
9.83  822 
9.83  849 
9.83  876 
9.83  903 
9.83  930 
9.83  957 

9.83  984 

9.84  011 
9.84  038 
9.84  065 
9.84092 
9.84 119 
9.84 146 
9.84  173 
9.84  200 
9.84  227 
9.84  254 
9.84  280 
9.84  307 
9.84  334 
9.84  361 
9.84  388 
9.84415 
9.84  442 
9.84  469 
9.84  496 
9.84  523 


0.17  101 
0.17  074 
0.17  047 
0.17  020 
0.16  992 
0.16  <)65 
0.16  938 
0.16  911 
0.16  883 
0.16  856 
0.16  829 
0.16  802 
0.16  775 
0.16  748 
0.16  720 
0.16  693 
0.16  666 
0.16  639 
0.16  612 
0.16  585 
0.16  558 
0.16  5:30 
0.16  503 
0.16  476 
0.16  449 
0.16  422 
0.16  395 
0.16  368 
0.16  341 
0.16  314 

0.16  287 
0.16  260 
0.16  232 
0.16  205 
0.16  178 
0.16  151 
0.16  124 
0.16  097 
0.16  070 
0.16  043 
0.16  016 
0.15  989 
0.15  962 
0.15  935 
0.15  908 
0.15  881 
0.15  854 
0.15  827 
0.15  800 
0.15  773 
0.15  746 
0.15  720 
0.15  693 
0.15  666 
0.15  639 
0.15  612 
0.15  585 
0.15  558 
0.15  531 
0.15  504 
0.15  477 


9.91 857 
9.91  849 
9.91  840 
9.91  832 
9.91  823 
9.91  815 
9.91  806 
9.91  798 
9.91  789 
9.91 781 
9.91  772 
9.91 763 
9.91 755 
9.91  746 
9.91  738 
9.91  729 
9.91  720 
9.91 712 
9.91  703 
9.91  695 
9.91  686 
9.91  677 
9.91 669 
9.91 660 
9.91  651 
9.91 643 
9.91  634 
9.91  625 
9.91  617 
9.91 608 
9.91  599 
9.91 591 
9.91  582 
9.91 573 
9.91  565 
9.91  556 
9.91  547 
9.91  538 
9.91 530 
9.91  521 
9.91 512 
9.91 504 
9.91 495 
9.91  486 
9.91  477 
9.91  469 
9.91 460 
9.91 451 
9.91 442 
9.91 433 
9.91 425 
9.91  416 
9.91 407 
9.91  398 
9.91  389 
9.91  381 
9.91  372 
9.91  363 
9.91  354 
9.91  345 
9.91  336 


60 

59 
58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

20 

19 
18 
17 
16 
15 
14 
13 
12 
11 
10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 


28 

27: 

2 

5.6 

5.4 

3 

8.4 

8.1 

4 

11.2 

10.8 

5 

14.0 

13.5 

6 

16.8 

16.2 

7 

19.6 

18.9 

8 

22.4 

21.6 

9 

25.2 

24.3 

26 

5.2 

7.8 
10.4 
13.0 
15.6 

18.2 
20.8 
23.4 


19 

18 

2 

3.8 

3.6 

3 

5.7 

5.4 

4 

7.6 

7.2 

5 

9.5 

9.0 

6 

11.4 

10.8 

7 

13.3 

12.6 

8 

15.2 

14.4 

9 

17.1 

16.2 

9 

2 

1.8 

3 

2.7 

4 

3.6 

5 

4.5 

6 

5.4 

7 

6.3 

8 

7.2 

9 

8.1 

1.6 
2.4 
3.2 
4.0 

4.8 
5.6 
6.4 

7.2 


From  the  top : 

For  34°+  or  214°+, 

read  as  printed  ;  for 
124°+  or  304°+,  read 
co-function. 

From  the  bottom : 

For  55°+  or  235°+, 
read  as  printed;  for 
145°+  or  325°+,  read 
co-function. 


L  Cos 


L  Gtn      c  d     L  Tan        L  Sin      d      ' 


Prop.  Pts. 


65°  —  Lofirarithms  of  Trisronomfttric  Functions 


35°  — logarithms  of  Trigonometric  Functions 


81 


L  Sin 


L  Tan 


c  d      L  Gtn 


L  Cos 


Prop.  Pts. 


0 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 

P7 
58 
59 
60 


9.75  859 
9.75  877 
9.75  895 
9.75  913 
9.75  931 
9.75  949 
9.75  967 

9.75  985 

9.76  003 
9.76  021 
9.76  039 
9.76  057 
9.76  075 
9.76  093 
9.76  111 
9.76  129 
9.76  146 
9.76  164 
9.76  182 
9.76  200 
9.76  218 
9.76  236 
9.76  253 
9.76  271 
9.76  289 
9.76  307 
9.76  324 
9.76  342 
9.76  360 
9.76  378 
9.76  395 
9.76  413 
9.76  431 
9.76  448 
9.76  466 
9.76  484 
9.76  501 
9.76  519 
9.76  537 
9.76  554 
9.76  572 
9.76  590 
9.76  607 
9.76  625 
9.76  642 
9.76  660 
9.76  677 
9.76  695 
9.76  712 
9.76  730 
9.76  747 
9.76  765 
9.76  782 
9.76  800 
9.76  817 
9.76  835 
9.76  852 
9.76  870 
9.76  887 
9.76  904 
9.76  922 


9.84  523 
9.84  550 
9.84  576 
9.84  603 
9.84  630 
9.84657 
9.84  684 
9.84  711 
9.84  738 
9.84  764 
9.84  791 
9.84  818 
9.84  845 
9.84  872 
9.84  899 
9.84  925 
9.84  952 

9.84  979 

9.85  006 
9.85  033 
9.85  059 
9.85  086 
9.85  113 
9.85  140 
9.85  166 
9.85  193 
9.85  220 
9.85  247 
9.85  273 
9.85  300 
9.85  327 
9.85  354 
9.85  380 
9.85  407 
9.85  434 
9.85  460 
9.85  487 
9.85  514 
9.85  540 
9.85  567 
9.85  594 
9.85  620 
9.85  647 
9.85  674 
9.85  700 
9.85  727 
9.85  754 
9.85  780 
9.85  807 
9.85  834 
9.85  860 
9.85  887 
9.85  913 
9.85  940 
9.85  967 

9.85  993 

9.86  020 
9.86  046 
9.86  073 
9.86  100 
9.86  136 


0.15  477 
0.15  450 
0.15  424 
0.15  397 
0.15  370 
0.15  343 
0.15  316 
0.15  289 
0.15  262 
0.15  236 
0.15  209 
0.15  182 
0.15  155 
0.15  128 
0.15  101 
0.15  075 
0.15  048 
0.15  021 
0.14  994 
0.14  967 
0.14  941 
0.14  914 
0.14  887 
0.14  8(K) 
0.14  834 
0.14807 
0.14  780 
0.14  753 
0.14  727 
0.14  700 
0.14  673 
0.14  646 
0.14  620 
0.14  593 
0.14  566 
0.14  540 
0.14  513 
0.14  486 
0.14  460 
0.14  433 
0.14  406 
0.14  380 
0.14  353 
0.14  326 
0.14  300 
0.14  273 
0.14  246 
0.14  220 
0.14 193 
0.14  166 
0.14 140 
0.14  113 
0.14  087 
0.14  060 
0.14  033 
0.14  007 
0.13  980 
0.13  954 
0.13  927 
0.13  900 
0.13  874 


9.91  336 
9.91  328 
9.91  319 
9.91  310 
9.91  301 
9.91 292 
9.91  283 
9.91  274 
9.91  266 
9.91 257 
9.91  248 
9.91 239 
9.91  230 
9.91  221 
9.91 212 
9.91  203 
9.91 194 
9.91 185 
9.91 176 
9.91 167 
9.91 158 
9.91 149 
9.91 141 
9.91 132 
9.91 123 
9.91 114 
9.91 105 
9.91 096 
9.91 087 
9.91  078 
9.91  069 
9.91  060 
9.91 051 
9.91 042 
9.91 033 
9.91 023 
9.91 014 
9.91 005 
9.90  996 
9.90  987 
9.90  978 
9.^)0  969 
9.f)0  960 
9.90  951 
9.90  942 
9.90  933 
9.90  924 
9.90  915 
9.90  906 
9.90  896 
9.90  887 
9.90  878 
9.90  869 
9.90  860 
9.90  851 
9.90  842 
9.90  832 
9.()0  823 
9.90  814 
9.90  805 
9.90  796 


27 

26 

2 

5.4 

5.2 

3 

8.1 

7.8 

4 

10.8 

10.4 

5 

13.5 

13.0 

6 

16.2 

15.6 

7 

18.9 

18.2 

8 

21.6 

20.8 

9 

24.3 

23.4 

18 

3.6 

6.4 

7.2 

9.0 

10.8 

12.6 

14.4 

16.2 


17 

2 

3.4 

3 

5.1 

4 

6.8 

5 

8.5 

6 

10.2 

7 

11.9 

8 

13.6 

9 

15.3 

9 

2 

1.8 

3 

2.7 

4 

3.6 

5 

4.5 

6 

5.4 

7 

6.3 

8 

7.2 

9 

8.1 

10 

2.0 
3.0 
4.0 
5.0 
6.0 
7.0 
8.0 
9.0 


1.6 
2.4 
3.2 
4.0 
4.8 
5.6 
6.4 
7.2 


From  the  top : 
For  35°+ or  215°+, 
read  as  printed ;  for 
125°+ or  305°+,  read 
co-function. 

From  the  bottom : 

For  54°+ or  234°+, 

read  as  printed ;  for 

144°+ or  324°+,  read 

co-function. 


L  Goa 


LCtn 


c  d      L  Tan 


L  Sin 


Prop.  Pts. 


54°— Logarithms  of  Trigonometric  Functions 


82 


36°  —  Logarithms  of  Trigonometric  Functions         [in 


L  Sin 


d   L  Tan  c  d   L  Ctn 


L  Cos 


Prop.  Pts. 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 


9.76  922 
9.76  939 
9.76  9r)7 
9.76  974 

9.76  991 

9.77  009 
9.77  026 
9.77  043 
9.77  061 
9.77  078 
9.77  095 
9.77  112 
9.77  130 
9.77  147 
9.77  164 
9.77  181 
9.77  199 
9.77  216 
9.77  233 
9.77  250 
9.77  268 
9.77  285 
9.77  302 
9.77  319 
9.77  336 
9.77  353 
9.77  370 
9.77  387 
9.77  405 
9.77  422 
9.77  439 
9.77  456 
9.77  473 
9.77  490 
9.77  507 
9.77  524 
9.77  541 
9.77  558 
9.77  575 
9.77  592 
9.77  609 
9.77  626 
9.77  643 
9.77  660 
9.77  677 
9.77  694 
9.77  711 
9.77  728 
9.77  744 
9.77  761 
9.77  778 
9.77  795 
9.77  812 
9.77  829 
9.77  846 
9.77  862 
9.77  879 
9.77  896 
9.77  913 
9.77  930 
9.77  946 


9.86  126 
9.86  153 
9.86  179 
9.86  206 
9.86  232 
9.86  259 
9.86  285 
9.86  312 
9.86  338 
9.86  365 
9.86  392 
9.86  418 
9.86  445 
9.86  471 
9.86  498 
9.86  524 
9.86  551 
9.86  577 
9.86  603 
9.86  630 
9.86  656 
9.86  683 
9.86  709 
9.86  736 
9.86  762 
9.86  789 
9.86  815 
9.86  842 
9.86  868 
9.86  894 
9.86  921 
9.86  947 

9.86  974 

9.87  000 
9.87  027 
9.87  053 
9.87  079 
9.87  106 
9.87  132 
9.87  158 
9.87  185 
9.87  211 
9.87  238 
9.87  264 
9.87  290 
9.87  317 
9.87  343 
9.87  369 
9.87  396 
9.87  422 
9.87  448 
9.87  475 
9.87  501 
9.87  527 
9.87  554 
9.87  580 
9.87  606 
9.87  633 
9.87  659 
9.87  685 
9.87  711 


27 

26 

27 

26 

27 

26 

27 

26 

27 

27 

26 

27 

26 

27 

26 

27 

26 

26 

27 

26 

27 

26 

27 

26 

27 

26 

27 

26 

26 

27 

26 

27 

26 

27 

26 

26 

27 

26 

26 

27 

26 

27 

26 

26 

27 

26 

26 

27 

26 

26 

27 

26 

26 

27 

26 

26 

27 

26 

26 

26 


0.13  874 
0.13  847 
0.13  821 
0.13  794 
0.13  768 
0.13  741 
0.13  715 
0.13  688 
0.13  662 
0.13  635 
0.13  608 
0.13  582 
0.13  555 
0.13  529 
0.13  502 
0.13  476 
0.13  449 
0.13  423 
0.13  397 
0.13  370 
0.13  344 
0.13  317 
0.13  291 
0.13  264 
0.13  238 
0.13  211 
0.13  185 
0.13  158 
0.13  132 
0.13106 
0.13  079 
0.13  053 
0.13  026 
0.13  000 
0.12  973 
0.12  947 
0.12  921 
0.12  894 
0.12  868 
0.12  842 
0.12  815 
0.12  789 
0.12  762 
0.12  736 
0.12  710 
0.12  683 
0.12  657 
0.12  631 
0.12  604 
0.12  578 
0.12  552 
0.12  525 
0.12  499 
0.12  473 
0.12  446 
0.12  420 
0.12  394 
0.12  367 
0.12  341 
0.12  315 
0.12  289 


9.90  796 
9.90  787 
9.90  777 
9.90  768 
9.90  759 
9.90  750 
9.90  741 
9.90  731 
9.90  722 
9.90  713 
9.90  704 
9.90  694 
9.90  685 
9.90  676 
9.90  667 
9.90  657 
9.90  648 
9.90  639 
9.90  630 
9.90  620 
9.90  611 
9.<)0  602 
9.90  592 
9.90  583 
9.90  574 
9.90  565 
9.90  555 
9.90  546 
9.90  537 
9.90  527 
9.90  518 
9.90  509 
9.90  499 
9.90  490 
9.90  480 
9.90  471 
9.90  462 
9.90  452 
9.90  443 
9.90434 
9.90  424 
9.90  415 
9.90  405 
9.90  396 
9.90  386 
9.90  377 
9.90  368 
9.90  358 
9.90349 
9.90  339 
9.90  330 
9.90320 
9.90  311 
9.90  301 
9.90  292 
9.90  282 
9.90  273 
9.90  263 
9.90  254 
9.90  244 
9.90  235 


60 

59 
58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 
10 
9 
8 
7 
6 
5 
4 
3 
2 
1 
0 


27 

26 

5.4 

5.2 

8.1 

7.8 

10.8 

10.4 

13.5 

13.0 

16.2 

15.6 

18.9 

18.2 

21.6 

20.8 

24.3 

23.4 

18 

3.6 

5.4 

7.2 

9.0 

10.8 

12.6 

14.4 

16.2 


17 

2 

3.4 

3 

5.1 

4 

6.8 

5 

8.5 

6 

10.2 

7 

11.9 

8 

13.6 

9 

15.3 

10 

2 

2.0 

3 

3.0 

4 

4.0 

5 

5.0 

6 

6.0 

7 

7.0 

8 

8.0 

9 

9.0 

16 

3.2 

4.8 

6.4 

8.0 

9.6 

11.2 

12.8 

14.4 


9 

1.8 
2.7 
3.6 
4.5 
5.4 
6.3 
7.2 
8.1 


From  the  top : 

For  36°+ or  216°+, 

read  as  printed ;  for 
126°+ or  306°+,  read 

co-function. 

From  the  bottom : 

For  53°+ or  233°+, 
read  as  printed;  for 
143°+  or  323°+,  read 
co-function. 


LGos 


L  Ctn    I  c  d 


L  Tan 


L  Sin       d      ' 


Prop.  Pts. 


63°— Logarithms  of  Trigonometric  Functions 


Ill] 


37°  —  Logarithms  of  Trigonometric  Functions 


83 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

30 

31 
32 
33 
34 
35 

m 

37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
■57 
58 
59 
60 


L  Sin 


9.77  946 
9.77  963 
9.77  980 

9.77  997 

9.78  013 
9.78  030 
9.78  047 
9.78  063 
9.78  080 
9.78  097 
9.78  113 
9.78  130 
9.78  147 
9.78  163 
9.78  180 
9.78  197 
9.78213 
9.78  230 
9.78  246 
9.78  263 
9.78  280 
9.78  296 
9.78  313 
9.78  329 
9.78  346 
9.78  362 
9.78  379 
9.78  395 
9.78  412 
9.78  428 
9.78  445 
9.78  461 
9.78  478 
9.78  494 
9.78  510 
9.78  527 
9.78  543 
9.78  560 
9.78  576 
9.78  592 
9.78  609 
9.78  625 
9.78  642 
9.78  658 
9.78  674 
9.78  691 
9.78  707 
9.78  723 
9.78  739 
9.78  756 

78  772 
78  788 
78  805 
78  821 
78  837 
78  853 
78  869 
78  886 
78  902 
78  918 
78  934 


L  Tan  c  d  L  Ctn 


9.87  711 
9.87  738 
9.87  764 
9.87  790 
9.87  817 
9.87  843 
9.87  869 
9.87  895 
9.87  922 
9.87  948 

9.87  974 

9.88  000 
9.88  027 
9.88  053 
9.88  079 
9.88  105 
9.88  131 
9.88  158 
9.88  184 
9.88  210 
9.88  236 
9.88  262 
9.88  289 
9.88  315 
9.88  341 
9.88  367 
9.88  393 
9.88  420 
9.88  446 
9.88  472 
9.88  498 
9.88  524 
9.88  550 
9.88  577 
9.88  603 
9.88  629 
9.88  655 
9.88  681 
9.88  707 
9.88  733 
9.88  759 
9.88  786 
9.88  812 
9.88  838 
9.88  864 
9.88  890 
9.88  916 
9.88  942 
9.88  968 

9.88  994 

9.89  020 
9.89  046 
9.89  073 
9.89099 
9.89 125 
9.89  151 
9.89  177 
9.89  203 
9.89  229 
9.89  255 
9.89  281 


0.12  289 
0.12  262 
0.12  236 
0.12  210 
0.12  183 
0.12  157 
0.12  131 
0.12105 
0.12  078 
0.12  052 
0.12  026 
0.12  000 
0.11  973 
0.11947 
0.11  921 
0.11  895 
0.11  869 
0.11  842 
0.11  816 
0.11  790 
0.11  764 
0.11  738 
0.11711 
0.11685 
0.11659 
0.11  633 
0.11 607 
0.11  580 
0.11  554 
0.11  528 
0.11  502 
0.11  476 
0.11 450 
0.11 423 
0.11  397 
0.11  371 
0.11  345 
0.11  319 
0.11  293 
0.11  267 
0.11241 
0.11214 
0.11 188 
0.11 162 
0.11 136 
0.11 110 
0.11  084 
0.11  058 
0.11  032 
0.11  006 
0.10  980 
0.10  954 
0.10  927 
0.10  901 
0.10  875 
0.10  849 
0.10  823 
0.10  797 
0.10  771 
0.10  745 
0.10  719 


L  Cos 


9.90  235 
9.90  225 
9.90  216 
9.W  206 
9.90  197 
9.90  187 
9.90  178 
9.90  168 
9.90  159 
9.90  149 
9.90  139 
9.90130 
9.90  120 
9.^)0  111 
9.90101 
9.90  091 
9.90  082 
9.90  072 
9.90  063 
9.90  053 
9.90  043 
9.90  034 
9i)0  024 
9.90  014 
9.90  005 
9.89  995 
9.89  985 
9.89  976 
9.89  966 
9.89  956 
9.89  947 
9.89  937 
9.89  927 
9.89  918 
9.89  908 
9.89  898 
9.89  888 
9.89  879 
9.89  869 
9.89  859 
9.89  849 
9.89  840 
9.89  830 
9.89  820 
9.89  810 
9.89  801 
9.89  791 
9.89  781 
9.89  771 
9.89  761 
9.89  752 
9.89  742 
9.89  732 
9.89  722 
9.89  712 
9.89  702 
9.89  693 
9.89  683 
9.89  673 
9.89  663 
9.89653  I 


Prop.  Pts. 


27 

26 

2 

5.4 

5.2 

3 

8.1 

7.8 

4 

10.8 

10.4 

5 

13.5 

13.0 

6 

16.2 

15.6 

7 

18.9 

18.2 

8 

21.6 

20.8 

9 

24.3 

23.4 

17 

3.4 

6.1 

6.8 

8.5 

10.2 

11.9 

13.6 

15.3 


9 

1.8 
2.7 
3.6 
4.5 
5.4 
6.3 
7.2 
8.1 


From  the  top : 

For37°+or217°+, 
read  as  printed;  for 
127°+ or  307°+,  read 
co-function. 

From  the  bottom  : 

For  52°+ or  232°+, 
read  as  printed  ;  for 
142°+  or  322°+,  read 
co-function. 


16 

10 

2 

3.2 

2.0 

3 

4.8 

3.0 

4 

6.4 

4.0 

6 

8.0 

5.0 

6 

9.6 

6.0 

7 

11.2 

7.0 

8 

12.8 

8.0 

9 

14.4 

9.0 

LGos 


L  Ctn     c  d 


LTan 


L  Sin 


Prop.  Pts. 


52°— Logarithms  of  Trigonometric  Functions 


84 


38°  — Logarithms  of  Trigonometric  Functions         [in 


L  Sin 


L  Tan     c  d      L  Ctn        L  Cos 


Prop.  Pts. 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 


27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


9.78  934 
9.78  950 
9.78  967 
9.78  983 

9.78  999 

9.79  015 
9.79  031 
9.79  047 
9.79063 
9.79  079 
9.79095 
9.79  111 
9.79  128 
9.79 144 
9.79 160 
9.79 176 
9.79192 
9.79  208 
9.79  224 
9.79  240 
9.79  256 
9.79272 
9.79  288 
9.79  304 
9.79  319 
9.79  335 
9.79  351 
9.79  367 
9.79  383 
9.79  399 
9.79415 
9.79431 
9.79  447 
9.79  463 
9.79478 
9.79494 
9.79  510 
9.79526 
9.79  542 
9.79  558 
9.79  573 
9.79589 
9.79605 
9.79  621 
9.79  636 
9.79652 
9.79668 
9.79  684 
9.79  699 
9.79  715 
9.79  731 
9.79  746 
9.79  762 
9.79  778 
9.79  793 
9.79  809 
9.79  825 
9.79  840 
9.79  856 
9.79  872 
9.79  887 


9.89  281 
9.89  307 
9.89  333 
9.89  359 
9.89  385 
9.89  411 
9.89  437 
9.89463 
9.89  489 
9.89  515 
9.89  541 
9.89  567 
9.89  593 
9.89  619 
9.89645 
9.89  671 
9.89  697 
9.89  723 
9.89  749 
9.89  775 
9.89  801 
9.89  827 
9.89  853 
9.89  879 
9.89  905 
9.89  931 
9.89  957 

9.89  983 
9.90009 

9.90  035 
9.90  061 
9.90  086 
9.90112 
9.90  138 
9.90  164 
9.90190 
9.90  216 
9.90  242 
9.90  268 
9.90  294 
9.90  320 
9.90  346 
9.90  371 
9.90  397 
9.90  423 
9.90  449 
9.90  475 
9.90  501 
9.90  527 
9.90  553 
9.90  578 
9.90  604 
9.90  630 
9.90  656 
9.90682 
9.90  708 
9m  734 
9.90  759 
9.90  785 
9.90  811 
9.90  837 


0.10  719 
0.10  693 
0.10  667 
0.10  641 
0.10  615 
0.10  589 
0.10  563 
0.10  537 
0.10  511 
0.10  485 
0.10459 
0.10  433 
0.10  407 
0.10  381 
0.10  355 
0.10  329 
0.10  303 
0.10  277 
0.10  251 
0.10  225 
0.10199 
0.10  173 
0.10147 
0.10121 
0.10  095 
0.10069 
0.10  043 
0.10  017 
0.09  991 
0.09  965 
0.09  939 
0.09  914 
0.09  888 
0.09  862 
0.09  836 
0.09  810 
0.09  784 
0.09  758 
0.09  732 
0.09  706 
0.09  680 
0.09  654 
0.09  629 
0.09  603 
0.09  577 
0.09  551 
0.09  525 
0.09499 
0.09  473 
0.09  447 
0.09  422 
0.09  396 
0.09  370 
0.09  344 
0.09  318 
0.09  292 
0.09  266 
0.09  241 
0.09  215 
0.09  189 
0.09 163 


9.89  653 
9.89  643 
9.89  633 
9.89  624 
9.89  614 
9.89  604 
9.89594 
9.89  584 
9.89574 
9.89  564 
9.89  554 
9.89  544 
9.89534 
9.89  524 
9.89514 
9.89504 
9.89495 
9.89  485 
9.89  475 
9.89465 
9.89  455 
9.89  445 
9.89  435 
9.89425 
9.89  415 
9.89405 
9.89  395 
9.89  385 
9.89  375 
9.89  364 
9.89  354 
9.89  344 
9.89  334 
9.89  324 
9.89  314 
9.89  304 
9.89  294 
9.89  284 
9.89  274 
9.89  264 
9.89  254 
9.89  244 
9.89  233 
9.89  223 
9.89  213 
9.89  203 
9.89  193 
9.89 183 
9.89 173 
9.89 162 
9,89152 
9.89 142 
9.89132 
9.89 122 
9.89112 
9.89  101 
9.89091 
9.89081 
9.89071 
9.89  060 
9.89  050 


60 

59 
58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 
10 


26 

25 

2 

5.2 

5.0 

3 

7.8 

7.5 

4 

10.4 

10.0 

5 

13.0 

12.5 

6 

15.6 

15.0 

7 

18.2 

17.5 

8 

20.8 

20.0 

9 

23.4 

22.5 

16 

15 

2 

3.2 

3.0 

3 

4.8 

4.5 

4 

6.4 

6.0 

5 

8.0 

7.5 

6 

9.6 

9.0 

7 

11.2 

10.5 

8 

12.8 

12.0 

9 

14.4 

13.5 

17 

3.4 

5.1 

6.8 

8.5 

10.2 

11.9 

13.6 

15.3 


11 

2.2 
'So 
4.4 
5.5 
6.() 
7.7 
8.8 
9.9 


10 

2 

2.0 

3 

3.0 

4 

4.0 

5 

5.0 

6 

6.0 

7 

7.0 

8 

8.0 

9 

9.0 

9 

1.8 

2.7 
3.6 
4.5 
5.4 
6.3 
7.2 
8.1 


From  the  top : 

For38°+or218°+, 

read  as  printed  ;  for 
128°+ or  308°+,  read 
co-function. 

From  the  bottom : 

For  51°+ or  231°+, 
read  as  printed;  for 
141°+  or  321°+,  read 
co-function. 


LOos 


L  Ctn      c  d     L  Tan 


L  Sin 


Prop.  Pts. 


51°  —  Logarithms  of  Trigonometric  Functions 


rii]         39°  —  Logarithms  of  Trigonometric  Functions 


S5 


L  Sin 


L  Tan     c  d      L  Ctn 


L  Cos 


Prop.  Pts. 


9.79  887 
9.79  903 
9.79  918 
9.79  934 
9.79  950 
9.79  %5 
9.79  981 

9.79  996 

9.80  012 
9.80  027 
9.80  043 
9.80  058 
9.80  074 
9.80  089 
9.80  105 
9.80  120 
9.80  136 
9.80  151 
9.80  166 
9.80  182 
9.80  197 
9.80  213 
9.80  228 
9.80  244 
9.80  259 
9.80  274 
9.80  290 
9.80  305 
9.80  320 
9.80  33() 
9.80  351 
9.80  366 
9.80  382 
9.80  397 
9.80  412 
9.80  428 
9.80  443 
9.80  458 
9.80  473 
9.80  489 
9.80  504 
9.80  519 
9.80  534 
9.80  550 
9.80  565 
9.80  580 
9.80  595 
9.80  610 
9.80  625 
9.80  641 
9.80  656 
9.80  671 
9.80  686 
9.80  701 
9.80  716 
9.80  731 
9.80  746 
9.80  762 
9.80  777 
9.80  792 
9.80  807 


9.90  837 
9.90  863 
9.90  889 
9.90  914 
9.90  940 
9.90  966 

9.90  992 
9.91 018 
9.91 043 

9.91  069 
9.91 095 
9.91  121 
9.91 147 
9.91 172 
9.91 198 
9.91  224 
9.91250 
9.91  276 
9.91  301 
9.91  327 
9.91  353 
9.91  379 
9.91 404 
9.91  430 
9.91 456 
9.91  482 
9.91 507 
9.91 533 
9.91  559 
9.91 585 
9.91  610 
9.91 636 
9.91  662 
9.91 688 
9.91 713 
9.91  739 
9.91  765 
9.91  791 
9.91  816 
9.91  842 
9.91  868 
9.91 893 
9.91  919 
9.91  945 

9.91  971 
9.91 996 

9.92  022 
9.92  048 
9.92  073 
9.92  099 
9.92  125 
9.92  150 
9.92  176 
9.92  202 
9.92  227 
9.92  253 
9.92  279 
9.92  304 
9.92  330 
9.92  356 
9.92  381 


0.09  163 
0.09 137 
0.09  111 
0.09  086 
0.09  060 
0.09034 
0.09  0C8 
0.08  982 
0.08  957 
0.08  931 
0.08  905 
0.08  879 
0.08  853 
0.08  828 
0.08  802 
0.08  776 
0.08  750 
0.08  724 
0.08  699 
0.08  673 
0.08  647 
0.08  621 
0.08  596 
0.08  570 
0.08  544 
0.08  518 
0.08  493 
0.08  467 
0.08  441 
0.08  415 
0.08  390 
0.08  364 
0.08  338 
0.08  312 
0.08  287 
0.08  261 
0.08  235 
0.08  209 
0.08  184 
0.08  158 
0.08  132 
0.08  107 
0.08  081 
0.08  055 
0.08  029 
0.08  004 
0.07  978 
0.07  952 
0.07  927 
0.07  901 
0.07  875 
0.07  850 
0.07  824 
0.07  798 
0.07  773 
0.07  747 
0.07  721 
0.07  696 
0.07  670 
0.07  644 
0.07  619 


9.89  050 
9.89  040 
9.89  030 
9.89  020 
9.89  009 
9.88  999 
9.88  989 
9.88  978 
9.88  968 
9.88  958 
9.88  948 
9.88  937 
9.88  927 
9.88  917 
9.88  906 
9.88  896 
9.88  886 
9.88  875 
9.88  865 
9.88  855 
9.88  844 
9.88  834 
9.88  824 
9.88  813 
9.88  803 
9.88  793 
9.88  782 
9.88  772 
9.88  761 
9.88  751 
9.88  741 
9.88  730 
9.88  720 
9.88  709 
9.88  699 
9.88  688 
9.88  678 
9.88  668 
9.88  657 
9.88  647 
9.88  636 
9.88  626 
9.88  615 
9.88  605 
9.88  594 
9.88  584 
9.88  573 
9.88  563 
9.88  552 
9.88  542 
9.88  531 
9.88  521 
9.88  510 
9.88  499 
9.88  489 
9.88  478 
9.88  468 
9.88  457 
9.88  447 
9.88  436 
9.88  425 


26 

25 

5.2 

5.0 

7.8 

7.5 

10.4 

10.0 

13.0 

12.5 

15.6 

15.0 

18.2 

17.5 

20.8 

20.0 

23.4 

22.5 

15 

11 

2 

3.0 

2.2 

3 

4.5 

3.3 

4 

6.0 

4.4 

5 

7.5 

5.5 

6 

9.0 

6.6 

7 

10.5 

7.7 

8 

12.0 

8.8 

9 

13.5 

9.9 

16 

3.2 

4.8 

6.4 

8.0 

9.6 

11.2 

12.8 

14.4 


10 

2.0 
3.0 
4.0 
5.0 
6.0 
7.0 
8.0 
9.0 


From  the  top : 

For  39°+ or  219°+, 
read  as  printed ;  for 
129°+ or  309°+,  read 
co-function. 

From  the  bottom  : 

For  60°+ or  230°+, 

read  as  printed ;  for 
140°+ or  320°+,  read 

co-function. 


LGos 


L  Ctn   c  d 


LTan 


L  Sin 


Prop.  Pts. 


50° — Logarithms  of  Trigonometric  Functions 


^6 


40*"  —  Logarithms  of  Trigonometric  Functions         [in 


LSin 


L  Tan     c  d      L  Ctn 


L  Cos 


Prop.  Pts. 


0 

1 
2 

3 
4 

5 

6 

7 

8 

9 

LO 

11 

12 

13 

14 

15 

m 

17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
3(3 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


9.80  807 
9.80  822 
9.80  837 
9.80  852 
9.80  867 
9.80  882 
9.80  897 
9.80  912 
9.80  927 
9.80  942 

9.80  957 
9.80  972 

9.80  987 

9.81  002 
9.81017 
9.81  032 
9.81  047 
9.81 061 
9.81 076 
9.81091 
9.81 106 
9.81 121 
9.81 136 
9.81 151 
9.81 166 
9.81 180 
9.81 195 
9.81  210 
9.81  225 
9.81  240 
9.81 254 
9.81  269 
9.81 284 
9.81  299 
9.81  314 
9.81 328 
9.81 343 
9.81  358 
9.81  372 
9.81 387 
9.81  402 
9.81 417 
9.81 431 
9.81  446 
9.81 461 
9.81  475 
9.81 490 
9.81 505 
9.81 519 
9.81  534 
9.81  549 
9.81  563 
9.81  578 
9.81 592 
9.81 607 
9.81  622 
9.81 636 
9.81  651 
9.81  665 
9.81  680 
9.81 694 


9.92  381 
9.92  407 
9.92  433 
9.92  458 
9.92  484 
9.92  510 
9.92  535 
9.92  561 
9.92  587 
9.92  612 

9.92  638 
9.92  663 
9.92  689 
9.92  715 
9.92  740 
9.92  766 
9.92  792 
9.92  817 
9.92  843 
9.92  868 
9.92  894 
9.92  920 
9.92  945 
9.92  971 

9.92  996 
9.93022 

9.93  048 
9.93  073 
9.93099 
9.93 124 
9.93 150 
9.93  175 
9.93  20r 
9.93  227 
9.93  252 
9.93  278 
9.93  303 
9.93  329 
9.93  354 
9.93  380 
9.93  406 
9.93  431 
9.93  457 
9.93482 
9.93508 
9.93  533 
9.93  559 
9.93  584 
9.93  610 
9.93  636 
9.93  661 
9.93  687 
9.93  712 
9.93  738 
9.93  763 
9.93  789 
9.93  814 
9.93  840 
9.93  865 
9.93  891 
9.93  916 


0.07  619 
0.07593 
0.07  567 
0.07  542 
0.07  516 
0.07  490 
0.07  465 
0.07  439 
0.07  413 
0.07  388 
0.07  362 
0.07  337 
0.07  311 
0.07  285 
0.07  260 
0.07  234 
0.07  208 
0.07  183 
0.07  157 
0.07  132 
0.07 106 
0.07  080 
0.07  055 
0.07  029 
0.07  004 
0.06  978 
0.06  952 
0.06  927 
0.06  901 
0.06  876 
0.06  850 
0.06  825 
0.06  799 
0.06  773 
0.06  748 
0.06  722 
0.06  697 
0.06  671 
0.06  646 
0.06620 
0.06  594 
0.06  569 
0.06  543 
0.06  518 
0.06  492 
0.06  467 
0.06  441 
0.06  416 
0.06  390 
0.06  364 
0.06  339 
0.06  313 
0.06  288 
0.06  262 
0.06  237 
0.06  211 
0.06  186 
0.06  160 
0.06  135 
0.06  109 
0.06  084 


9.88  425 
9.88  415 
9.88  404 
9.88  394 
9.88  383 
9.88  372 
9.88  362 
9.88  351 
9.88  340 
9.88  330 
9.88  319 
9.88  308 
9.88  298 
9.88  287 
9.88  276 
9.88  266 
9.88  255 
9.88  244 
9.88  234 
9.88  223 
9.88  212 
9.88  201 
9.88  191 
9.88  180 
9.88  169 
9.88  158 
9.88  148 
9.88 137 
9.88  126 
9.88 115 
9.88  105 
9.88  094 
9.88  083 
9.88  072 
9.88  061 
9.88051 
9.88  040 
9.88  029 
9.88  018 
9.88007 
9.87  996 
9.87  985 
9.87  975 
9.87  964 
9.87  953 
9.87  942 
9.87  931 
9.87  920 
9.87  909 
9.87  898 
9.87  887 
9.87  877 
9.87  866 
9.87  855 
9.87  844 
9.87833 
9.87  822 
9.87  811 
9.87  800 
9.87  789 
9.87  778 


26 

25 

2 

6.2 

5.0 

3 

7.8 

7.5 

4 

10.4 

10.0 

5 

13.0 

12.5 

6 

15.6 

15.0 

7 

18.2 

17.5 

8 

20.8 

20.0 

9 

23.4 

22.5 

14 

11 

2.8 

2.2 

4.2 

3.3 

5.6 

4.4 

7.0 

5.5 

8.4 

6.6 

9.8 

7.7 

11.2 

8.8 

12.6 

9.9 

15 

3.0 

4.5 

6.0 

7.5 

9.0 

10.5 

12.0 

13.5 


10 

2.0 
3.0 
4.0 
5.0 
6.0 
7.0 
8.0 
9.0 


From  the  top  : 

For40<^+or220°+, 

read  as  printed ;  for 
130°+ or  310°+,  read 

co-function.  ^ 

From  the  bottom : 

For  49°+ or  229°+, 
read  as  printed;  for 
139°+ or  319°+,  read 
co-function. 


LGos 


LCtn 


c  d     L  Tan 


LSin 


Prop.  Pts. 


49°  — Logarithms  of  Trigonometric  Functions 


ill] 


41°  —  Logarithms  of  Trigonometric  Functions 


LSin 


L  Tan     c  d     L  Ctn 


L  Cos 


Prop.  Pts. 


10 

11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
-57 
58 
59 
60 


9.81 694 
9.81  709 
9.81  723 
9.81  738 
9.81  752 
9.81  767 
9.81  781 
9,81  796 
9.81  810 
9.81  825 
9.81  839 
9.81  854 
9.81  868 
9.81  882 
9.81  897 
9.81  911 
9.81  926 
9.81  940 
9.81  955 
9.81  969 
9.81  983 

9.81  998 

9.82  012 
9.82  026 
9.82  041 
9.82  055 
9.82  069 
9.82  084 
9.82  01)8 
9.82  112 
9.82  126 
9.82  141 
9.82  155 
9.82  169 
9.82  184 
9.82  198 
9.82  212 
9.82  226 
9.82  240 
9.82  255 
9.82  269 
9.82  283 
9.82  297 
9.82  311 
9.82  326 
9.82  340 
9.82  354 
9.82  368 
9.82  382 
9.82  396 
9.82  410 
9.82  424 
9.82  439 
9.82  453 
9.82  467 
9.82  481 
9.82  495 
9.82  509 
9.82  523 
9.82  537 
9.82  551 


L  Cos 


9.93  916 
9.93  942 
9.93  967 

9.93  993 
9.94018 

9.94  044 
9.94  069 
9.94  095 
9.94 120 
9.94 146 
9.94171 
9.94  197 
9.94  222 
9.94  248 
9.94  273 
9.94  299 
9.94  324 
9.94  350 
9.94  375 
9.94401 
9.94426 
9.94452 
9.94  477 
9.94  503 
9.94528 
9.94  554 
9.94  579 
9.94  604 
9.94  630 
9.94  655 
9.94  681 
9.94  706 
9.94  732 
9.94  757 
9.94  783 
9.94  808 
9.94  834 
9.94  859 
9.94  884 
9.94  910 
9.94  935 
9.94  961 

9.94  986 

9.95  012 
9.95  037 
9.95  062 
9.95  088 
9.95  113 
9.95  139 
9.95  164 
9.95  190 
9.95  215 
9.95  240 
9.95  266 
9.95  291 
9.95  317 
9.95  342 
9.95  368 
9.95  393 
9.95  418 
9.95  444 


L  Ctn     c  d 


0.06  084 
0.06  058 
0.06033 
0.06  007 
0.05  982 
0.05  956 
0.05  931 
0.05  905 
0.05  880 
0.05  854 
0.05  829 
0.05  803 
0.05  778 
0.05  752 
0.05  727 
0.05  701 
0.05  676 
0.05  650 
0.05  625 
0.05  599 
0.05  574 
0.05  548 
0.05  523 
0.05  497 
0.05472 
0.05  446 
0.05  421 
0.05  396 
0.05  370 
0.05  345 
0.05  319 
0.05  294 
0.05  268 
0.05  243 
0.05  217 
0.05  192 
0.05  166 
0.05  141 
0.05  116 
0.05  090 
0.05  065 
0.05  039 
0.05  014 
0.04  988 
0.04  963 
0.04  938 
0.04  912 
0.04  887 
0.04  861 
0.04  836 
0.04  810 
0.04  785 
0.04  760 
0.04  734 
0.04  709 
0.04  683 
0.04  658 
0.04  632 
0.04  607 
0.04  582 
0.04  556 


9.87  778 
9.87  767 
9.87  756 
9.87  745 
9.87  734 
9.87  723 
9.87  712 
9.87  701 
9.87  690 
9.87  679 
9.87  668 
9.87  657 
9.87  646 
9.87  635 
9.87  624 
9.87  613 
9.87  601 
9.87  590 
9.87  579 
9.87  668 
9.87  557 
9.87  546 
9.87  635 
9.87  524 
9.87  613 
9.87  601 
9.87  490 
9.87  479 
9.87  468 
9.87  457 
9.87  446 
9.87  434 
9.87  423 
9.87  412 
9.87401 
9.87  3C0 
9.87  378 
9.87  367 
9.87  356 
9.87  345 
9.87  334 
9.87  322 
9.87  311 
9.87  300 
9.87  288 
9.87  277 
9.87  266 
9.87  255 
9.87  243 
9.87  232 
9.87  221 
9.87  209 
9.87  198 
9.87  187 
9.87  175 
9.87  164 
9.87  153 
9.87  141 
9.87  130 
9.87  119 
9.87  107 


26 

25 

2 

6.2 

5.0 

3 

7.8 

7.5 

4 

10.4 

10.0 

5 

13.0 

12.5 

6 

15.6 

15.0 

7 

18.2 

17.5 

8 

20.8 

20.0 

9 

23.4 

22.5 

14 

12 

2 

2.8 

2.4 

3 

4.2 

3.6 

4 

6.6 

4.8 

6 

7.0 

6.0 

6 

8.4 

7.2 

7 

9.8 

8.4 

8 

11.2 

9.6 

9 

12.6 

10.8 

15 

3.0 

4.6 

6.0 

7.5 

9.0 

10.5 

12.0 

13.5 


11 

2.2 
3.3 
4.4 
5.5 
6.6 
7.7 
8.8 
9.9 


From  the  top : 

For41°+or221^+, 
read  as  printed;  for 
131°+ or  311°+,  read 
co-function. 

From  the  bottom : 

For  48°+ or  228°+, 
read  as  printed  ;  for 
138°+ or  318°+,  read 
co-function. 


L  Tan 


L  Sin 


Prop.  Pts. 


48° — Logarithms  of  Trigonometric  Functions 


88 


42°  —  Logarithms  of  Trigonometric  Functions         [iii 


L  Sin 


L  Tan  c  d  L  Ctn 


L  Cos 


Prop.  Pts. 


10 

11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


9.82  551 
9.82  565 
9.82  579 
9.82  593 
9.82  607 
9.82  621 
9.82  635 
9.82  649 
9.82  663 
9.82  677 
9.82  691 
9.82  705 
9.82  719 
9.82  733 
9.82  747 
9.82  761 
9.82  775 
9.82  788 
9.82  802 
9.82  816 
9.82  830 
9.82  844 
9.82  858 
9.82  872 
9.82  885 
9.82  899 
9.82  913 
9.82  927 
9.82  941 
9.82  955 

9.82  968 
9.82  982 

9.82  996 
9.83010 
9.83023 

9.83037 
9.83051 

9.83  065 
9.83  078 
9.83092 
9.83106 
9.83 120 
9.83 133 
9.83 147 
9.83 161 
9.83 174 
9.83 188 
9.83  202 
9.83  215 
9.83  229 
9, 
9, 
9, 


83  242 
83  256 
83  270 
83  283 
83  297 
83  310 
83  324 
83  338 
83  351 
.83  365 
.83  378 


9.95  444 
9.95  469 
9.95  495 
9.95  520 
9.95  545 
9.95  571 
9.95  596 
9.95  622 
9.95  647 
9.95  672 
9.95  698 
9.95  723 
9.95  748 
9.95  774 
9.95  799 
9.95  825 
9.95  850 
9.95  875 
9.95  901 
9.95  926 
9.95  952 

9.95  977 
9.96002 

9.96  028 
9.96053 
9.96  078 
9.96104 
9.96 129 
9.96  155 
9.96180 
9.96  205 
9.96  231 
9.96  256 
9.96  281 
9.96  307 
9.96  332 
9.96  357 
9.96  383 
9.96  408 
9.96  433 
9.96  459 
9.96  484 
9.96  510 
9.96  535 
9.96  560 
9.96  586 
9.96  611 
9.96  636 
9.96  662 
9.96  687 
9.9f)712 
9.96  738 
9.96  763 
9.96  788 
9.96  814 
9.9i]  839 
9.t)6  8<>4 
9.96  890 
9.96  915 
9.96  940 
9.96  966 


0.04  556 
0.04  531 
0.04  505 
0.04  480 
0.04  455 
0.04  429 
0.04404 
0.04  378 
0.04  353 
0.04  328 
0.04  302 
0.04  277 
0.04  252 
0.04  226 
0.04  201 
0.04 175 
0.04  150 
0.04  125 
0.04  099 
0.04  074 
0.04  048 
0.04  023 
0.03  998 
0.03  972 
0.03  947 
0.03  922 
0.03  896 
0.03  871 
0.03  845 
0.03  820 
0.03  795 
0.03  769 
0.03  744 
0.03  719 
0.03  693 
0.03  668 
0.03  643 
0.03  617 
0.03  592 
0.03  567 
0.03  541 
0.03  516 
0.03  490 
0.03  465 
0.03440 
0.03  414 
0.03  389 
0.03  364 
0.03  338 
0.03  313 
0.03  288 
0.03  262 
0.03  237 
0.03  212 
0.03  186 
0.03 161 
0.03 136 
0.03  110 
0.03085 
0.03  060 
0.03  034 


9.87  107 
9.87  096 
9.87  085 
9.87  073 
9.87  062 
9.87  050 
9.87  039 
9.87  028 
9.87  016 
9.87  005 
9.86  993 
9.86  982 
9.86  970 
9.86  959 
9.86  947 
9.86  936 
9.86  924 
9.86  913 
9.86  902 
9.86890 
9.86  879 
9.86  867 
9.86  855 
9.86  844 
9.86  832 
9.86  821 
9.86  809 
9.86  798 
9.86  786 
9.86  775 
9.86  763 
9.86  752 
9.86  740 
9.86  728 
9.86  717 
9.86  705 
9.86  694 
9.86  682 
9.86  670 
9.86659 
9.86  647 
9.86  635 
9.86  624 
9.86  612 
9.86  600 
9.86  589 
9.86  577 
9.86  565 
9.86  554 
9.86  542 
9.86  530 
9.86  518 
9.86  507 
9.86  495 
9.86  483 
9.86  472 
9.86  460 
9.86  448 
9.86  436 
9.86425 
9.86  413 


26 

25 

2 

5.2 

5.0 

3 

7.8 

7.5 

4 

10.4 

10.0 

5 

13.0 

12.5 

6 

15.6 

15.0 

7 

18.2 

17.5 

8 

20.8 

20.0 

9 

23.4 

22.5 

13 

12 

2 

2.6 

2.4 

3 

3.9 

3.6 

4 

5.2 

4.8 

5 

6.5 

6.0 

6 

7.8 

7.2 

7 

9.1 

8.4 

8 

10.4 

9.6 

9 

11.7 

10.8 

14 

2.8 
4.2 
5.6 
7.0 
8.4 
9.8 
11.2 
12.6 


11 

2.2 
8.3 
4.4 
5.5 
6.6 
7.7 
8.8 
9.9 


From  the  top: 

For42°+or222'^+, 
read  as  printed ;  for 


132°+ or  312°+ 

co-function. 


read 


From  the  bottom : 

ror47°+or227°+, 
read  as  printed;  for 
137°+ or  317°+,  read 
co-function. 


L  Cos 


L  Ctn      c  d 


L  Tan 


L  Sin 


Prop.  Pts. 


47°  — Logarithms  of  Trigonometric  Functions 


Ill] 


43^  —  Logarithms  of  Trigonometric  Functions 


89 


L  Sin 


L  Tan 


c  d      L  Gtn 


L  Cos       d 


Prop.  Pts. 


3 
4 
5 
(3 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 

45 

46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
'57 
58 
59 
60 


9.83  378 
9.83  392 
9.83  405 
9.83  419 
9.83  432 
9.83  446 
9.83  459 
9.83  473 
9.83  486 
9.83  500 
9.83513 
9.83  527 
9.83  540 
9.83  554 
9.83  567 
9.83  581 
9.83  594 
9.83  608 
9.83  621 
9  83  634 
9.83  648 
9.83  661 
9.83  674 
9.83  688 
9.83  701 
9.83  715 
9.83  728 
9.83  741 
9.83  755 
9.83  768 
9.83  781 
9.83  795 
9.83  808 
9.83  821 
9.83  834 
9.83  848 
9.83  861 
9.83  874 
9.83  887 
9.83  901 
9 
9, 
9 
9, 
9 
9, 
9, 
9 
9, 
9 


83  914 
.83  927 

83  940 
.83  954 

.83  967 

83  980 

83  993 

84  006 
84  020 
.84  033 

9.84  046 
9.84  059 
9.84  072 
9.84  085 
9.84  098 
9.84112 
9.84  125 
9.84  138 
9.84  151 
9.84  164 
9.84  177 


9.96  966 

9.96  991 

9.97  016 
9.97  042 
9.97  067 
9.97  092 
9.97  118 
9.97  143 
9.97  168 
9.97  193 
9.97  219 
9.97  244 
9.97  269 
9.97  295 
9.97  320 
9.97  345 
9.97  371 
9.97  396 
9.97  421 
9.97  447 
9.97  472 
9.97  497 
9.97  523 
9.97  548 
9.97  573 
9.97  598 
9.97  624 
9.97  649 
9.97  674 
9.97  700 
9.97  725 
9.97  750 
9.97  776 
9.97  801 
9.97  826 
9.97  851 
9.97  877 
9.97  902 
9.97  927 
9.97  953 

9.97  978 

9.98  003 
9.98  029 
9.98  054 
9.98  079 
9.98  104 
9.98  130 
9.98  155 
9.98  180 
9.98  206 
9.98  231 
9.98  256 
9.98  281 
9.98  307 
9.98  332 
9.98  357 
9.98  383 
9.98  408 
9.98  433 
9.98  458 
9.98  484 


0.03  034 
0.03009 
0.02  984 
0.02  958 
0.02  933 
0.02  908 
0.02  882 
0.02  857 
0.02  832 
0.02  807 
0.02  781 
0.02  756 
0.02  731 
0.02  705 
0.02  680 
0.02  655 
0.02  629 
0.02  604 
0.02  579 
0.02  553 
0.02  528 
0.02  503 
0.02  477 
0.02  452 
0.02  427 
0  02  402 
0.02  376 
G.02  351 
0.02  326 
0.02  300 
0.02  275 
0.02  250 
0.02  224 
0.02  199 
0.02  174 
0.02  149 
0.02  123 
0.02  098 
0.02  073 
0.02  047 
0.02  022 
0.01  997 
0.01  971 
0.01  946 
0.01  921 
0.01  896 
0.01  870 
0.01  845 
0.01  820 
0.01  794 
0.01  769 
0.01  744 
0.01  719 
0.01  693 

0.01  ms 

0.01  643 
0.01  617 
0.01  592 
0.01  567 
0.01  542 
0.01  516 


9.86  413 
9.86  401 
9.86  389 
9.86  377 
9.86  366 
9.8()  354 
9.86  342 
9.86  330 
9.86  318 
9.86  306 
9.86  295 
9.86  283 
9.86  271 
9.86  259 
9.86  247 
9.86  235 
9.8()  223 
9.86  211 
9.86  200 
9.86  188 
9.8(3 176 
9.86  164 
9.86  152 
9.86  140 
9.86  128 
9.86  116 
9.86  104 
9.86  092 
9.86  080 
9.86  068 
9.86  056 
9.86  044 
9.86  032 
9.86  020 
9.86  008 
9.85  996 
9.85  984 
9.85  972 
9.85  960 
9.85  948 
9.85  93(3 
9.85  924 
9.85912 
9.85^)00 
9.85  888 
9.85  876 
9.85  864 
9.85  851 
9.85  839 
8.85  827 
9.85  815 
9.85  803 
9.85  791 
9.85  779 
9.85  766 
9.85  754 
9.85  742 
9.85  730 
9.85  718 
9.85  706 
9.85  693 


26 

25 

2 

5.2 

5.0 

3 

7.8 

7.5 

4 

10.4 

10.0 

5 

13.0 

12.5 

6 

15.6 

15.0 

7 

18.2 

17.5 

8 

.20.8 

20.0 

9 

23.4 

22.5 

13 

12 

2 

2.6 

2.4 

3 

3.9 

3.6 

4 

5.2 

4.8 

5 

6.5 

6.0 

H 

7.8 

7.2 

7 

9.1 

8.4 

8 

10.4 

9.6 

9 

11.7 

10.8  1 

14 

2.8 
4.2 
5.6 
7.0 
8.4 
9.8 
11.2 
12.6 


11 

2.2 
3.3 
4.4 
5.5 
6.6 
7.7 
8.8 
9.9 


From  the  top  : 

For  43°+ or  223°+, 
read  as  printed ;  for 
133°+ or  313°+,  read 
co-function. 

Frojii  the  bottom  : 

For  46°+ or  226°+, 
read  as  printed ;  for 
136°+ or  316°+,  read 
co-function. 


LCos 


LCtn 


c  d      L  Tan 


LSin    Id 


Prop.  Pts. 


46°— Losrarithms  of  Trigonometric  Functions 


90 


44°  —  Logarithms  of  Trigonometric  Functions         [in 


10 

11 

12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
3f) 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


LSin 


9.84177 
9.84  190 
9.84  203 
9.84  216 
9.84  229 
9.84  242 
9.84  255 
9.84  269 
9.84  282 
9.84  295 
9.84  308 
9.84  321 
9.84  334 
9.84  347 
9.84  360 
9.84  373 
9.84  385 
9.84  398 
9.84  411 
9.84  424 
9.84  437 
9.84  450 
9.84  463 
9.84  476 
9.84  489 
9.84  502 
9.84  515 
9.84  528 
9.84  540 
9.84  553 
9.84  566 
9.84  579 
9.84  592 
9.84  605 
9.84  618 
9.84  630 
9.84  643 
9.84  656 
9.84  669 
9.84  682 
9.84  694 
9.84  707 
9.84  720 
9.84  733 
9.84  745 
9.84  758 
9.84  771 
9.84  784 
9.84  796 
9.84  809 
9.84  822 
9.84  835 
9.84  847 
9.84  860 
9.84  873 
9.84  885 
9.84  898 
9.84  911 
9.84  923 
9.84  936 
9.84  949 


LGos 


L  Tan 


9.98  484 
9.98  509 
9.98  534 
9.98  560 
9.98  585 
9.98  610 
9.98  635 
9.98  661 
9.98  686 
9.98  711 
9.98  737 
9.98  762 
9.98  787 
9.98  812 
9.98  838 
9.98  863 
9.98  888 
9.98  913 
9.98  939 
9.98  964 

9.98  989 

9.99  015 
9.99  040 
9.99065 
9.99  090 
9.99 116 
9.99 141 
9.99  166 
9.99  191 
9.99  217 
9.99  242 
9.99  267 
9.99  293 
9.99  318 
9.99  343 
9.99  368 
9.99  394 
9.99  419 
9.99444 
9.99  469 
9.99  495 
9.99  520 
9.99  545 
9.99  570 
9.99  596 
9.99  621 
9.99  ()46 
9.99  672 
9.99  697 
9.99  722 
9.99  747 
9.99  773 
9.99  798 
9.99  823 
9.99  848 
9.99  874 
9.99  899 
9.99  924 
9.99  949 
9.99  975 
0.00  000 


LCtn 


c  d   L  Ctn 


0.01  516 
0.01 491 
0.01  466 
0.01 440 
0.01 415 
0.01  390 
0.01  365 
0.01  339 
0.01  314 
0.01  289 
0.01  263 
0.01  2.38 
0.01  213 
0.01 188 
0.01 162 
0.01 137 
0.01 112 
0.01  087 
0.01 061 
0.01036 
0.01011 
0.00  985 
0.00  960 
0.00  935 
0.00  910 
0.00  884 
0.00  859 
0.00  834 
0.00  809 
0.00  783 
0.00  758 
0.00  733 
0.00  707 
0.00  682 
0.00  657 
0.00  632 
0.00  606 
0.00  581 
0.00  556 
0.00  531 
0.00  505 
0.00  480 
0.00  455 
0.00430 
0.00404 
0.00  379 
0.00  354 
0.00  328 
0.00  303 
0.00  278 
0.00  253 
0.00  227 
0.00  202 
0.00  177 
0.00  152 
0.00  126 
0.00  101 
0.00  076 
0.00  051 
0.00  025 
0.00  000 


c  d     L  Tan 


LGos 


9.85  693 
9.85  681 
9.85  669 
9.85  657 
9.85  645 
9.85  632 
9.85  620 
9.85  608 
9.85  596 
9.85  583 
9.85  571 
9.85  559 
9.85  547 
9.85  534 
9.85  522 
9.85  510 
9.85  497 
9.85  485 
9.85  473 
9.85  460 
9.85  448 
9.85  436 
9.85  423 
9.85  411 
9.85  399 
9.85  386 
9.85  374 
9.85  361 
9.85  349 
9.85  337 
9.85  324 
9.85  312 
9.85  299 
9.85  287 
9.85  274 
9.85  262 
9.85  250 
9.85  237 
9.85  225 
9.85  212 
9.85  200 
9.85  187 
9.85  175 
9.85  162 
9.85  150 
9.85  1.37 
9.85  125 
9.85  112 
9.85  100 
9.85  087 
9.85  074 
9.85  062 
9.85  049 
9.85  037 
9.85  024 
9.85  012 
9.84  999 
9.84  986 
9.84  974 
9.84  961 
9  84949 


L  Sin 


Prop.  Pts. 


26 

25 

2 

5.2 

5.0 

3 

7.8 

7.5 

4 

10.4 

10.0 

5 

13.0 

12.5 

6 

15.6 

15.0 

7 

18.2 

17.5 

8 

20.8 

20.0 

9 

23.4 

22.5 

14 

2.8 
4.2 
5.6 
7.0 
8.4 
9.8 
11.2 
12.6 


13 

2- 

2.6 

3 

3.9 

4 

5.2 

5 

6.5 

6 

7.8 

7 

9.1 

8 

10.4 

9 

11.7 

12 

2.4 
3.6 

4.8 
6.0 
7.2 
8.4 
9.6 
10.8 


From  the  top  : 

For  44°+ or  224°+, 
read  as  printed;  for 
134°+ or  314°+,  read 
co-function. 

Fro7n  the  bottom  : 

For  45°+ or  225°+, 
read  as  printed ;  for 
135°+ or  315°+,  read 
co-function. 


Prop.  Pts. 


45°  —  Logarithms  of  Triaronometric  Functions 


IV] 

Table  IV  — Degrees, 

Minutes,  and  Seconds  to  Radians   91 

Degrees 

Minutes 

Seconds 

0° 

0.0000000 

60° 

1.04719  76 

120° 

2.09439  51 

0' 

0.00000  00 

0" 

0.00000  00 

1 

0.01745  33 

61 

1.0(;465  08 

121 

2.11184  84 

1 

0.00029  09 

1 

0.00000  48 

2 

0.03190  66 

62 

1.08210  41 

122 

2.12930  17 

2 

0.00058  18 

2 

0.00000  97 

3 

0.05235  99 

63 

1.09955  74 

123 

2.14675  50 

3 

0.00087  27 

3 

0.00001 45 

4 

0.06981  32 

64 

1.11701  07 

124 

2.16420  83 

4 

0.00116  36 

4 

0.00001  94 

5 

0.08726  65 

65 

1.13446  40 

125 

2.18166  16 

5 

0.00145  44 

5 

0.00002  42 

6 

0.1047198 

66 

1.1519173 

126 

2.19911  49 

6 

0.00174  53 

6 

0.00002  91 

7 

0.12217  30 

67 

1.16937  06 

127 

2.21656  82 

7 

0.00203  62 

7 

0.00003  39 

8 

0.13962  63 

68 

1.18682  39 

128 

2.23402  14 

8 

0.00232  71 

8 

0.00003  88 

9 

0.15707  96 

69 

1.20427  72 

129 

2.25147  47 

9 

0.00261  80 

9 

0.00004  36 

10 

0.17453  29 

70 

1.22173  05 

130 

2.26892  80 

10 

0.00290  89 

10 

0.00004  85 

11 

0.19198  62 

71 

1.23918  38 

131 

2.28638  13 

11 

0.00319  98 

11 

0.00005  33 

12 

0.20943  95 

72 

1.25663  71 

132 

2.3038346 

12 

0.00349  07 

12 

0.00005  82 

13 

0.22689  28 

73 

1.27409  04 

133 

2.32128  79 

13 

0.00378  15 

13 

0.00006  30 

14 

0.24434  61 

74 

1.29154  36 

134 

2.33874 12 

14 

0.00407  24 

14 

0.00006  79 

15 

0.26179  94 

75 

1.30899  69 

135 

2.35619  45 

15 

0.00436  33 

15 

0.00007  27 

16 

0.27925  27 

76 

1.32645  02 

136 

2.37364  78 

16 

0.00465  42 

16 

0.00007  76 

17 

0.29670  60 

77 

1.31390  35 

137 

2.39110  11 

17 

0.00494  51 

17 

0-00008  24 

18 

0.31415  93 

78 

1.36135  68 

138 

2.40855  44 

18 

0.00523  60 

18 

0.00008  73 

19 

0.33161 26 

79 

1.3788101 

139 

2.42600  77 

19 

0.00552  69 

19 

0.00009  21 

20 

0.34906  59 

80 

1.39626  34 

140 

2.44346  10 

20 

0.00581  78 

20 

0.00009  70 

21 

0.36651  91 

81 

1.4137167 

141 

2.46091  42 

21 

0.00610  87 

21 

0.00010 18 

22 

0.38397  24 

82 

1.4.3117  00 

142 

2.47836  75 

22 

0.00639  95 

22 

0.00010  67 

23 

0.40142  57 

83 

1.44862  33 

143 

2.49582  08 

23 

0.00669  04 

23 

0.00011 15 

24 

0.41887  90 

84 

1.46607  66 

144 

2.51327  41 

24 

0.00698  13 

24 

0.0001164 

25 

0.43633  23 

85 

1.48352  99 

145 

2.53072  74 

25 

0.00727  22 

25 

0.00012 12 

26 

0.45378  56 

86 

1.50098  32 

146 

2.54818  07 

26 

0.00756  31 

26 

0.00012  61 

27 

0.47123  89 

87 

1.51843  64 

147 

2.56563  40 

27 

0.00785  40 

27 

0.00013  09 

28 

0.48869  22 

88 

1.53588  97 

148 

2.58308  73 

28 

0.a)814  49 

28 

0.00013  57 

29 

0.50614  55 

89 

1.55334  30 

149 

2.60054  06 

29 

0.00843  58 

29 

0.00014  06 

30 

0.52359  88 

90 

1.57079  63 

150 

2.61799  39 

30 

0.00872  66 

30 

0.00014  54 

31 

0.54105  21 

91 

1.58824  9(5 

151 

2.63544  72 

31 

0.00901  75 

31 

0.00015  03 

32 

0.55850  54 

92 

1.60570  29 

152 

2.65290  05 

32 

0.00930  84 

32 

0.00015  51 

33 

0.57595  87 

93 

1.62315  62 

153 

2.67035  38 

33 

0.00959  93 

33 

0.00016  00 

34 

0.59341 19 

94 

1.64060  95 

154 

2.68780  70 

34 

0.00989  02 

34 

0.00016  48 

35 

0.61086  52 

95 

1.65806  28 

155 

2.70526  03 

35 

0.01018  11 

35 

0.00016  97 

36 

0.62831  85 

96 

1.67551  61 

156 

2.72271  36 

36 

0.01047  20 

36 

0.00017  45 

37 

0.64577  18 

97 

1.69296  94 

157 

2.74016  69 

37 

0.01076  29 

37 

0.00017  94 

38 

0.66322  51 

98 

1.71042  27 

158 

2.75762  02 

38 

0.01105  38 

38 

0.00018  42 

39 

0.68067  84 

99 

1.72787  60 

159 

2.77507  35 

39 

0.01134  46 

39 

0.00018  91 

40 

0.69813  17 

100 

1.74532  93 

160 

2.79252  68 

40 

0.01163  55 

40 

0.00019  39 

41 

0.71558  50 

101 

1.76278  25 

161 

2.80998  01 

41 

0.01192  64 

41 

0.00019  88 

42 

0.73303  83 

102 

1.78023  58 

162 

2.82743  34 

42 

0.01221  73 

42 

0.00020  36 

43 

0.75049 16 

103 

1.79768  91 

163 

2.84488  67 

43 

0.01250  82 

43 

0.00020  85 

44 

0.76794  49 

104 

1.81514  24 

164 

2.86234  00 

44 

0.01279  91 

44 

0.00021  33 

45 

0.78530  82 

105 

1.83259  57 

165 

2.87979  33 

45 

0.01309  00 

45 

0.00021  82 

46 

0.8028515 

106 

1.8.5004  90 

166 

2.89724  66 

46 

0.01338  09 

46 

0.00022  30 

47 

0.82030  47 

107 

1.86750  23 

167 

2.91469  99 

47 

0.01367  17 

47 

0.00022  79 

48 

0.83775  80 

108 

1.88495  56 

168 

2.93215  31 

48 

0.01.S96  26 

48 

0.00023  27 

49 

0.85521 13 

109 

1.90240  89 

169 

2.94960  64 

49 

0.01425  35 

49 

0.00023  76 

50 

0.87266  46 

110 

1.91986  22 

170 

2.96705  97 

50 

0.01454  44 

50 

0.00024  24 

51 

0.89011  79 

111 

1.9373155 

171 

2.98451  30 

51 

0.01483  53 

51 

0.00024  73 

52 

0.90757  12 

112 

1.95476  88 

172 

3.00196  63 

52 

0.01512  62 

52 

0.00025  21 

53 

0.92502  45 

113 

1.97222  21 

173 

3.01941  96 

53 

0.01541  71 

53 

0.00025  70 

54 

0.94247  78 

114 

1.98967  53 

174 

3.03687  29 

54 

0.01570  80 

54 

0.00026  18 

55 

0.95993  11 

115 

2.00712  86 

175 

3.05432  62 

55 

0.01599  89 

55 

0.00026  66 

56 

0.97738  44 

116 

2.0245819 

176 

3.07177  95 

56 

0.01628  97 

56 

0.00027  15 

57 

0.99483  77 

117 

2.04203  52 

177 

3.08923  28 

57 

0.01658  06 

57 

0.00027  63 

58 

1.0122910 

118 

2.05948  85 

178 

3.10668  61 

58 

0.01687  15 

58 

0.00028 12 

59 

1.02974  43 

119 

2.07694  18 

179 

3.12413  94 

59 

0.01716  24 

59 

0.00028  60 

60 

1.04719  76 

120 

2.09439  51 

180 

3.14159  27 

60 

0.01745  33 

60 

0.00029  09 

92 


V— Radian  Measure  —  Trigonometric  Functions         [t 


i 

8 

Sin  x 

Cos  a? 

Tana; 

I. 

.00 

.00000 

1.0000 

.00000 

0°00'.0 

.01 
.02 
.03 

.04 
.05 
.06 

.07 

.08 
.09 

.01000 
.02000 
.03000 

.03999 
.04998 
.05996 

.06994 
.07991 
.08988 

.999^5 
.99980 
.99955 

.99920 

.99875 
.99820 

.99755 
.99680 
.99595 

.01000 
.02000 
.03001 

.04002 
.05004 
.06007 

.07011 
.08017 
.09024 

0°34'.4 
1°08'.8 
l°43'.l 

2°  17'. 5 
2°51'.9 
3°  26'. 3 

4°  00'. 6 
4°.35'.0 
5°  09'. 4 

.10 

.09983 

.99500 

.10033 

5°  43'. 8 

.11 
.12 
.13 

.14 
.15 
.16 

.17 

.18 
.19 

.10978 
.11971 
.12963 

.13954 
.14944 
.15932 

.16918 
.17903 

.18886 

.99396 
.99281 
.99156 

.99022 

.98877 
.98723 

.98558 

.98384 
.98200 

.11045 
.12058 
.13074 

.14092 
.15114 
.16138 

.17166 
.18197 
.19232 

6°18'.2 
6°  52'. 5 
7°26'.9 

8°  01'. 3 
8°35'.7 
9°10'.0 

9°  44'. 4 

10°  18'. 8 
10°  53'. 2 

.20 

.19867 

.98007 

.20271 

11°  27' .5 

.21 
.22 
.23 

.24 
.25 
.26 

.27 
.28 
.29 

.20846 
.21823 
.22798 

.23770 
.24740 
.25708 

.26673 
.27636 
.28595 

.97803 
.97590 
.97367 

.97134 

.96891 
.96639 

.96377 
.96106 
.95824 

.21314 
.22362 
.23414 

.24472 
.25534 
.26602 

.27676' 

.28755 
.29841 

12°01'.9 
12°  36'. 3 
13°10'.7 

13°45'.l 
14°  19' .4 
14°  53'. 8 

15°28'.2 
16°  02'. 6 
16°36'.9 

.30 

.29552 

.95534 

.30934 

17°11'.3 

.31 
.32 
.33 

.34 

.35 
.36 

.37 
.38 
.39 

.30506 
.31457 
.32404 

.33349 
.34290 
.35227 

.36162 
.37092 
.38019 

.95233 
.94924 
.94604 

.94275 
.93937 
.93590 

.93233 
.92866 
.92491 

.32033 
.33139 
.34252 

.35374 
.36503 
.37640 

.38786 
.39941 
.41106 

17°45'.7 
18°20'.l 
18°  54' .5 

19°  28'. 8 
20°  03'.  2 
20°37'.6 

21°12'.0 
21°  46'. 3 

22°  20'. 7 

.40 

.38942 

.92106 

.42279 

22°55'.l 

.41 
.42 
.43 

.44 
.45 
.46 

.47 
.48 
.49 

.39861 
.40776 
.41687 

.42594 
.43497 
.44395 

.45289 
.46178 
.47063 

.91712 
.91309 
.90897 

.90475 
.90045 
.89605 

.89157 
.88699 
.88233 

.43463 
.44657 
.45862 

.47078 
.48305 
.49545 

.50795 
.52061 
.53339 

23°  29' .5 
24°  03'. 9 

24°  38'. 2 

25°12'.6 
25°47'.0 
26°  21'. 4 

26°55'.7 
27°30'.l 
28°  04' .5 

.60 

.47943 

.87758 

.54630 

28°  38'. 9 

CO 

1 

8 

Sin  a; 

Cos  a; 

Tana; 

1 

.50 

.47943 

.87758 

.54630 

28°  38'. 9 

.51 
.52 
.53 

.54 
.55 
.56 

.57 
.58 
.59 

.48818 
.49688 
.50553 

.51414 
.52269 
.53119 

.53963 
.54802 
.55636 

.87274 
.86782 
.86281 

.85771 
.85252 
.84726 

.84190 
.83646 
83094 

.55936 
.57256 
.58592 

.59943 
.61311 
.62695 

.64097 
.65517 
.66956 

29°  13'. 3 
29°  47'. 6 
30°22'.0 

30°  56'. 4 
31°  30'. 8 
32°05'.l 

32°  39'. 5 
33°  13'. 9 
33°  48'. 3 

.60 

.56464 

.82534 

.68414 

34°  22'. 6 

.61 

.62 
.63 

.64 
.65 
.66 

.67 
.68 
.69 

.57287 
.58104 
.58914 

.59720 
.60519 
.61312 

.62099 
.62879 
.63654 

.81965 
.81388 
.80803 

.80210 
.79608 
.78999 

.78382 
.77757 
.77125 

.69892 
.71391 
.72911 

.74454 
.76020 
.77610 

.79225 
.80866 
.82533 

34°57'.0 
35°  31 '.4 

36°  05'.8 

36°  40'. 2 
37°  14'. 5 

37°  48'. 9 

38°  23'. 3 

38°57'.7 
39°32'.0 

.70 

.64422 

.76484 

.84229 

40°  06'. 4 

.71 
.72 
.73 

.74 

.75 
.76 

.77 
.78 
.79 

.65183 
.65938 
.66687 

.67429 
.68164 
.68892 

.69614 
.70328 
.71035 

.75836 
.75181 
.74517 

.73847 
.73169 

.72484 

.71791 
.71091 

.70385 

.85953 
.87707 
.89492 

.91309 
.93160 
.95055 

.96967 
.98926 
1.0092 

40°  40'. 8 
41°15'.2 
41°  49'. 6 

42°  23'. 9 

42°58'.3 
43°  32'. 7 

44°07'.l 
44°41'.4 
45°  15'. 8 

.80 

.71736 

.69671 

1.0296 

45°50'.2 

.81 
.82 
.83 

.84 
.85 
.86 

.87 
.88 
.89 

.72429 
.73115 
.73793 

.74464 

.75128 
.75784 

.76433 
.77074 

.77707 

.68950 
.68222 
.67488 

.66746 
.65998 
.65244 

.64483 
.63715 
.62941 

1.0505 
1.0717 
1.0934 

1.1156 
1.1383 
1.1616 

1.1853 
1.2097 
1.2346 

46°  24' .6 
46°59'.0 
47°  33'. 3 

48°  07'. 7 
48°42'.l 
49°  16'. 5 

49°50'.8 
50°  25'. 2 
50°  59'. 6 

.90 

.78333 

.62161 

1.2602 

51°34'.0 

.91 
.92 
.93 

.94 
.95 
.96 

.97 

.98 
.99 

.78950 
.79560 
.80162 

.80756 
.81342 
.81919 

.82489 
.83050 
.83603 

.61375 
.60582 
.59783 

.58979 
.58168 
.57352 

.56530 
.55702 
.54869 

1.2864 
1.3133 
1.3409 

1.3692 
1.3984 
1.4284 

1.4592 
1.4910 
1.5237 

52°08'.3 
52°  42'. 7 
53°17'.l 

53°51'.5 
54°  25'. 9 
55°  00'. 2 

55°  34'. 6 
56°09'.0 
56°  43'. 4 

1.00 

.84147 

.54030 

1.5574 

57°17'.7 

V — Radian  Measure  —  Trigonometric  Functions        93 


8 

Sin  a; 

Cos  a? 

Tana; 

1 

1.00 

.84147 

.54030 

1.5574 

57°17'.7 

1.01 
1.02 
1.03 

1.04 
1.05 
1.06 

1.07 
1.08 
1.09 

.84683 
.85211 
.85730 

.86240 
.86742 

.87236 

.87720 
.88196 
.88663 

.53186 
.52337 
.51482 

.50622 
.49757 

.48887 

.48012 
.47133 
.46249 

1.5922 
1.6281 
1.6652 

1.7036 
1.7433 
1.7844 

1.8270 
1.8712 
1.9171 

57°52'.l 

58°  26'. 5 
59°  00'. 9 

59°  35'. 3 
60°  09'. 6 
60°  44'.0 

61°18'.4 
61°52'.8 
62°27'.l 

1.10 

.89121 

.45360 

l.<)648 

63°  01'. 5 

1.11 
1.12 
1.13 

1.14 
1.15 
1.16 

1.17 
1.18 
1.19 

.89570 
.90010 
.90441 

.90863 
.91276 
.91680 

.92075 
.92461 
.92837 

.44466 
.43568 
.42666 

.41759 
.40849 
.39934 

.39015 
.38092 
.37166 

2.0143 
2.0660 
2.1198 

2.1759 
2.2345 
2.2958 

2.3600 
2.4273 
2.4979 

63°  35'. 9 
64°  10'. 3 
64°  44'. 7 

65°19'.0 
65°  53'. 4 
66°  27'.8 

67°  02' .2 
67°  36'. 5 
68°  10'. 9 

1.20 

.93204 

.36236 

2.5722 

68°  45'. 3 

1.21 
1.22 
1.23 

1.24 
1.25 
1.26 

1.27 

1.28 
1.29 

.93562 
.93910 
.94249 

.94578 
.94898 
.95209 

.95510 
.95802 
.96084 

.35302 
.34365 
.33424 

.32480 
.31532 
.30582 

.29628 
.28672 
.27712 

2.6503 
2.7328 
2.8198 

2.9119 
3.0096 
3.1133 

3.2236 
3.3413 
3.4672 

69°  19'. 7 
69°54'.l 

70°  28'. 4 

71°  02'. 8 
71°  37'. 2 
72°  11'. 6 

72°  45'. 9 
73°  20'. 3 

73°  54'. 7 

1.30 

.96356 

.26750 

3.6021 

74°29'.l 

1 

8 

Sin  a; 

Cos  a? 

Tana; 

1 

r 

1.30 

.96356 

.26750 

3.6021 

74°29'.l 

1.31 
1.32 
1.33 

1.34 
1.35 
1.36 

1.37 
1.38 
1.39 

.96618 
.90872 
.97115 

.97348 
.97572 

.97786 

.97991 
.98185 
.98370 

.25785 
.24818 
.23848 

.22875 
.21901 
.20924 

.19945 
.18964 
.17981 

3.7470 
3.9033 
4.0723 

4.2556 

4.4552 
4.6734 

4.9131 
5.1774 

5.4707 

75°  03'. 4 

75°  37 '.8 
76°  12'. 2 

76°4(r.6 
77°21'.0 
77°  55'. 3 

78°29'.7 
79°04'.l 
79°  38'. 5 

1.40 

.98545 

.16997 

5.7979 

80°  12'. 8 

1.41 
1.42 
1.43 

1.44 
1.45 
1.46 

1.47 
1.48 
1.49 

.98710 
.98865 
.99010 

.99146 
.99271 
.99387 

.99492 
.99588 
.99674 

.16010 
.15023 
.14033 

.13042 
.12050 
.11057 

.10063 
.09067 
.08071 

6.16.^4 
6.5811 
7.0555 

7.6018 
8.2381 
8.9886 

9.8874 
10.983 
12.350 

80°47'.2 
81°21'.6 
81°56'.0 

82°  30'. 4 
83°  04'. 7 
83°39'.l 

84°  13'. 5 
84°47'.9 
85°  22'. 2 

1.50 

.99749 

.07074 

14.101 

85°  56'. 6 

1.51 
1.52 
1.53 

1.54 
1.55 
1.56 

1.57 

1.58 
1.59 

.99815 
.99871 
.99917 

.99953 
.99978 
.99994 

1.0000 
.999% 
.99982 

.06076 
.05077 
.04079 

.03079 
.02079 
.01080 

.00080 
-.00920 
-.01920 

16.428 
19.670 
24.498 

32.461 
48.078 
92.621 

1255.8 
-108.65 
-52.067 

86°31'.0 
87°  05'. 4 
87°  39'. 8 

88°14'.l 
88°  48'. 5 
89°  22'. 9 

89°  57'. 3 
90°  31 '.6 
91°06'.0 

1.60 

.99957 

-.02920 

-34.233 

91°40'.4 

TT  radians  =  180°  ir  =  3.14159265 

1  radian  =  57°  17'  44".806  =  57.°  2957795 

3600"  =  60'  =  1°  =  .01745329  radian 


TABLE  V  a  — RADIANS   TO  DEGREES 


Radians 

Tenths 

Hundredths 

Thousandths 

Ten-thousandths 

1 

57°17'44".8 

5°43'46".5 

0°34'22".6 

0°  3'26".3 

0°  0'20".6 

2 

114°35'29".6 

11°27'33".0 

1°  8'45".3 

0°  6'52".5 

0°  0'41".3 

3 

171°53'14".4 

17°11'19".4 

1°43'07".9 

0°10'18".8 

0°  1'01".9 

4 

229°10'59".2 

22°55'05".9 

2°17'30".6 

0°13'45".l 

0°  1'22".5 

5 

286°28'44".0 

28°38'52".4 

2°51'53".2 

0°17'11".3 

0°  l'43".l 

6 

343°46'28".8 

34°22'38".9 

3°26'15".9 

0°20'37".6 

0°  2'03".8 

7 

401°  4' 13"  .6 

40°  6'25".4 

4°  0'38".5 

0°24'03".9 

0°  2'24".4 

8 

458°21'58".4 

45°50'11".8 

4°35'01".2 

0°27'30".l 

0°  2'45".0 

9 

515°39'43".3 

61°33'58".3 

5°  9'23".8 

0°30'56".4 

0°  3'05".6 

94 

Table  VI- 

-  Powers  —  Roots  —  Reciprocals 

[VI 

n 

n2 

Vn 

VlOn 

n^ 

^ 

^10  n 

1/n 

^100  li 

1.00 

1.0000 

1.00000 

3.16228 

1.00000 

1.00000 

2.15443 

4.64159 

1.00000 

1.01 
1.02 
1.03 

1.04 
1.05 
1.06 

1.07 

1.08 
1.09 

1.0201 
1.0404 
1.0609 

1.0816 
1.1025 
1.1236 

1.1449 
1.1664 
1.1881 

1.00499 
1.00995 
1.01489 

1.01980 
1.02470 
1.02956 

1.03441 
1.03923 
1.04403 

3.17805 
3.19374 
3.20936 

3.22490 
3.24037 
3.25576 

3.27109 
3.28634 
3.30151 

1.03030 
1.06121 
1.09273 

1.12486 
1.15762 
1.19102 

1.22504 
1.25971 
1.29503 

1.00332 
1.00662 
1.00990 

1.01316 
1.01640 
1.01961 

1.02281 
1.02599 
1.02914 

2.16159 
2.16870 
2.17577 

2.18279 
2.18976 
2.19669 

2.20358 
2.21042 
2.21722 

4.65701 
4.67233 
4.68755 

4.70267 
4.71769 
4.73262 

4.74746 
4.76220 

4.77686 

.9<)0099 
.980392 
.970874 

.961538 
.952381 
.943396 

.934579 
.925926 
.917431 

1.10 

1.2100 

1.04881 

3.31662 

1.33100 

1.03228 

2.22398 

4.79142 

.90fX)91 

1.11 
1.12 
1.13 

1.14 
1.15 
1.16 

1.17 
1.18 
1.19 

1.2321 
1.2544 
1.2769 

1.2996 
1.3225 
1.3456 

1.3689 
1.3924 
1.4161 

1.05357 
1.05830 
1.06:301 

1.06771 
1.07238 
1.07703 

1.08167 
1.08628 
1.09087 

3.33167 
3.34664 
3.36155 

3.37639 
3.39116 
3.40588 

3.42053 
3.43511 
3.44964 

1.36763 
1.40493 
1.44290 

1.48154 

1.52088 
1.56090 

1.60161 
1.64303 
1.68516 

1.03540 
1.03850 
1.04158 

1.04464 
1.04769 
1.05072 

1.05373 
1.05672 
1.05970 

2.23070 
2.23738 
2.24402 

2.25062 
2.25718 
2.26370 

2.27019 
2.27664 
2.28305 

4.80590 
4.82028 
4.83459 

4.84881 
4.86294 
4.87700 

4.89097 
4.90487 
4.91868 

.900901 
.892857 
.884956 

.877193 
.869565 
.862069 

.854701 
.847458 
.840336 

1.20 

1.4400 

1.09545 

3.46410 

1.72800 

1.06266 

2.28943 

4.93242 

.833333 

1.21 
1.22 
1.23 

1.24 
1.25 
1.26 

1.27 
1.28 
1.29 

1.4641 
1.4884 
1.5129 

1.5376 
1.5625 
1.5876 

1.6129 
1.6384 
1.6641 

1.10000 
1.10454 
1.10905 

1.11355 
1.11803 
1.12250 

1.12694 
1.13137 
1.13578 

3.47851 
3.49285 
3.50714 

3.52136 
3.53553 
3.54965 

3.56371 
3.57771 
3.59166 

1.77356 
1.81585 
1.86087 

1.90662 
1.95312 
2.00038 

2.04838 
2.09715 
2.14669 

1.06560 
1.06853 
1.07144 

1.07434 
1.07722 
1.08008 

1.08293 
1.08577 
1.08859 

2.29577 
2.30208 
2.30835 

2.31459 
2.32079 
2.32697 

2.33311 
2.33921 
2.34529 

4.94609 
4.95968 
4.97319 

4.98663 
5.00000 
6.01330 

5.02653 
5.03968 
5.05277 

.826446 
.819672 
.813008 

.806452 
.800000 
.793651 

.787402 
.781250 
.775194 

1.30 

1.6900 

1.14018 

3.60555 

2.19700 

1.09139 

2.35133 

5.06580 

.769231 

1.31 
1.32 
1.33 

1.34 
1.35 
1.36 

1.37 
1.38 
1.39 

1.7161 
1.7424 
1.7689 

1.7956 
1.8225 
1.8496 

1.8769 
1.9044 
1.9321 

1.14455 
1.14891 
1.15326 

1.15758 
1.16190 
1.16619 

1.17047 
1.17473 
1.17898 

3.61939 
3.63318 
3.64692 

3.66060 
3.67423 
3.68782 

3.70135 
3.71484 
3.72827 

2.24809 
2.29997 
2.35264 

2.40610 
2.46038 
2.51546 

2.57135 

2.62807 
2.68562 

1.09418 
1.096% 
1.09972 

1.10247 
1.10521 
1.10793 

1.11064 
1.11334 
1.11602 

2.35735 
2.36333 
2.36928 

2.37521 
2.38110 
2.38697 

2.39280 
2.39861 
2.40439 

5.07875 
5.09164 
6.10447 

5.11723 
5.12993 
5.14256 

5.15514 
5.16765 
5.18010 

.763359 
.757576 
.751880 

.746269 
.740741 
.735294 

.729927 
.724638 
.719424 

1.40 

1.9600 

1.18322 

3.74166 

2.74400 

1.11869 

2.41014 

5.19249 

.714286 

1.41 
1.42 
1.43 

1.44 
1.45 
1.46 

1.47 
1.48 
1.49 

1.9881 
2.0164 
2.0449 

2.0736 
2.1025 
2.1316 

2.1609 
2.1904 
2.2201 

1.18743 
1.19164 
1.19583 

1.20000 
1.20416 
1.20830 

1.21244 
1.21655 
1.22066 

3.75500 
3.76829 
3.78153 

3.79473 

3.80789 
3.82099 

3.83406 
3.84708 
3.86005 

2.80322 
2.86329 
2.92421 

2.98598 
3.04862 
3.11214 

3.17652 
3.24179 
3.30795 

1.12135 
1.12399 
1.12662 

1.12924 
1.13185 
1.13445 

1.13703 
1.13960 
1.14216 

2.41587 
2.42156 
2.42724 

2.43288 
2.43850 
2.44409 

2.44966 
2.45520 
2.46072 

5.20483 
5.21710 
5.22932 

5.24148 
5.25359 
6.26564 

5.27763 
5.28957 
5.30146 

.709220 
.704225 
.699301 

.694444 
.689655 
.684932 

.680272 
.675676 
.671141 

1.50 

2.2500 

1.22474 

3.87298 

3.37500 

1.14471 

2.46621 

5.31329 

.666667 

n 

n2 

Vw^ 

VlOn 

n^ 

^n 

^10  w. 

1/n 

^100  n 

VI] 

Powers  — Roots  — 

Reciprocals 

95 

n 

n^ 

\/n 

VlOtfc 

n^ 

^n 

^10  w, 

1/n 

</imn 

1.50 

2.2500 

1.22474 

3.87298 

3.37500 

1.14471 

2.4(5(521 

5.31329 

.666667 

1.51 
1.52 
1.53 

1.54 
1.55 
1.56 

1.57 
1.58 
1.59 

2.2801 
2.3104 
2.3409 

2.3716 
2.4025 
2.4336 

2.4649 
2.4964 
2.5281 

1.22882 
1^23288 
1.23693 

1.24097 
1.24499 
1.24900 

1.25300 
1.25698 
1.26095 

3.88587 
3.89872 
3.91152 

3.92428 
3.93700 
3.94968 

3.96232 
3.97492 
3.98748 

3.44295 
3.51181 
3.58158 

3.65226 
3.72388 
3.79642 

3.86989 
3.94431 
4.01968 

1.14725 
1.14978 
1.15230 

1.15480 
1.15J29 
1.15978 

1.16225 
1.1()471 
1.16717 

2.47168 
2.47712 
2.48255 

2.48794 
2.49332 
2.49867 

2.50399 
2.50930 
2.51458 

5.32507 
5.33680 
5.34»48 

5.36011 
5.37169" 
5.38321 

5.39469 
5.40612 
5.41750 

.662252 
.657895 
.653595 

.649351 
.645161 
.641026 

.6.36943 
.632911 
.628931 

1.60 

2.5600 

1.26491 

4.00000 

4.09600 

1.16961 

2.51984 

5.42884 

.625000 

1.61 
1.62 
1.63 

1.64 
1  .()5 
1.66 

1.67 
1.68 
1.69 

2.5921 
2.6244 
2.6569 

2.6896 
2.7225 
2.7556 

2.7889 
2.8224 
2.8561 

1.26886 
1.27279 
1.27671 

1.28062 
1.28452 
1.28841 

1.29228 
1.29615 
1.30000 

4.01248 
4.02492 
4.03733 

4.04969 
4.06202 
4.07431 

4.08656 

4.09878 
4.11096 

4.17328 
4.25153 
4.33075 

4.41094 
4.49212 
4.57430 

4.65746 
4.74163 

4.82681 

1.17204 
1.17446 
1.17687 

1.17927 
1.181(57 
1.18405 

1.18642 
1.18878 
1.19114 

2.52508 
2.53030 
2.53549 

2.54067 
2.54582 
2.55095 

2.55607 
2.56116 
2.56623 

5.44012 
5.45136 
5.46256 

5.47370 

5.48481 
5.49586 

5.50688 
5.51785 

5.52877 

.621118 
.617284 
.613497 

.609756 
.(506061 
.602410 

.598802 
.595238 
.591716 

1.70 

2.8900 

1.30384 

4.12311 

4.91300 

1.19348 

2.57128 

5.53966 

.588235 

1.71 
1.72 
1.73 

1.74 
1.75 
1.76 

1.77 
1.78 
1.79 

2.9241 

2.9584 
2.9929 

3.0276 
3.0625 
3.0976 

3.1329 
3.1684 
3.2041 

1.30767 
1.31149 
1.31529 

1.31909 
1.32288 
1.32665 

1.33041 
1.33417 
1.33791 

4.13521 
4.14729 
4.15933 

4.17133 
4.18330 
.4.19524 

4.20714 
4.2U)00 
4.23084 

5.00021 
5.08845 
5.17772 

5.26802 
5.35938 
5.45178 

5.54523 
5.(53975 
5.73534 

1.19582 
1.19815 
1.20046 

1.20277 
1.20507 
1.20736 

1.20964 
1.21192 
1.21418 

2.57631 
2.58133 
2.58632 

2.59129 
2.59625 
2.60118 

2.60610 
2.61100 
2.61588 

5.55050 
5.56130 
5.57205 

5.58277 
5.59344 
5.60408 

5.61467 
5.62523 
5.63574 

.584795 
.581.395 
.578035 

.574713 
.571429 

.568182 

.564972 
.561798 
.558659 

1.80 

3.2400 

1..34164 

4.24264 

5.83200 

1.21644 

2.62074 

5.64622 

.555556 

1.81 
1.82 
1.83 

1.84 
1.85 
1.86 

1.87 
1.88 
1.89 

3.2761 
3.3124 
3.3489 

3.3856 
3.4225 
3.4596 

3.4969 
3.5344 
3.5721 

1.3453(5 
1.34907 
1.35277 

1.35647 
1.36015 
1.36382 

1.36748 
1.37113 
1.37477 

4.25441 
4.26615 

4.27785 

4.28952 
4.30116 
4.31277 

4.32435 
4.335^K) 
4.34741 

5.92974 
6.02857 
6.12849 

6.22950 
6.33162 
6.43486 

6.53920 
6.64467 
6.75127 

1.21869 
1.22093 
1.22316 

1.22539 
1.22760 
1.22981 

1.23201 
1.23420 
1.23639 

2.(52559 
2.63041 
2.63522 

2.64001 
2.64479 
2.64954 

2.65428 
2.65901 
2.66371 

5.65(565 
5.66705 
5.67741 

5.68773 
5.69802 
5.70827 

5.71848 
5.72865 
5.73879 

.552486 
.549451 
.546448 

.543478 
.540541 
.537634 

.534759 
.531915 
.529101 

1.90 

3.6100 

1.37840 

4.35890 

6.85900 

1.23856 

2.66840 

5.74890 

.526316 

1.91 
1.92 
1.93 

1.94 
1.95 
1.96 

1.97 

.  1.98 

1.99 

3.6481 
3.6864 
3.7249 

3.7636 
3.8025 
3.8416 

3.8809 
3.9204 
3.9601 

1.38203 
1.38564 
1.38924 

1.39284 
1.39(542 
1.40000 

1.40357 
1.40712 
1.41067 

4.37035 
4.38178 
4.39318 

4.40454 
4.41588 
4.42719 

4.43847 
4.44972 
4.46094 

6.96787 
7.07789 
7.18906 

7.30138 
7.41488 
7.52954 

7.64537 
7.76239 
7.88060 

1.24073 
1.24289 
1.24.505 

1.24719 
1.24933 
1.25146 

1.25359 
1.25571 
1.25782 

2.67307 
2.67773 
2.68237 

2.68700 
2.69161 
2.69620 

2.70078 
2.70534 
2.70989 

5.75897 
5.76900 
5.77900 

5.78896 
5.79889 
5.80879 

5.81865 

5.82848 
5.83827 

.523560 
.520833 
.518135 

.5154(54 
.512821 
.510204 

.507614 
.505051 
.502513 

2.00 

4.0000 

1.41421 

4.47214 

8.00000 

1.25992 

2.71442 

5.84804 

.500000 

n 

n^ 

^n 

VlOw, 

n^ 

^n 

^10  n 

1/n 

^100  n 

96 

Powers  — Roots  — 

-  Recipi 

•ocals 

Vn 

n 

,  W2 

Vn 

VlOn 

n^ 

i^n 

^10^ 

l/n 

VlOOn 

2.00 

4.0000 

1.41421 

4.47214 

8.00000 

1.25992 

2.71442 

5.84804 

.500000 

2.01 
2.02 
2.03 

2.04 
2.05 
2.06 

2.07 
2.08 
2.09 

4.0401 
4.0804 
4.1209 

4.1616 
4.2025 
4.2436 

4.2849 
4.3264 
4.3681 

1.41774 
1.42127 
1.42478 

1.42829 
1.43178 
1.43527 

1.43875 
1.44222 
1.44568 

4  48330 
4.49444 
4.50555 

4.51664 
4.5^769 

4.53872 

4.54973 
4.56070 
4.57165 

8.12060 
8.24241 
8.36543 

8.48966 
8.61512 
8.74182 

8.86974 
8.99891 
9.12933 

1.26202 
1.26411 
1.26619 

1.26827 
1.27033 
1.27240 

1.27445 
1.27650 
1.27854 

2.71893 
2.72344 
2.72792 

2.73239 

2.73685 
2.74129 

2.74572 
2.75014 
2.75454 

5.85777 
5.86746 
5.87713 

5.88677 
5.89637 
5.90594 

5.91548 
5.92499 
5.93447 

.497512 
.495050 
.492611 

.490196 

.487805 
.485437 

.483092 
.480769 
.478469 

2.10 

4.4100 

1.44914 

4.58258 

9  26100 

1.28058 

2.75892 

5.94392 

.476190 

2.11 
2.12 
2.13 

2.14 
2.15 
2.16 

2.17 
2.18 
2.19 

4.4521 
4.4944 
4.5369 

4.5796 
4.6225 
4.6656 

4.7089 
4.7524 
4.7961 

1.45258 
1.45602 
1.45945 

1.46287 
1.46629 
1.46969 

1.47309 
1.47648 
1.47986 

4.59347 
4.60435 
4.61519 

4.62601 
4.63681 
4.64758 

4.65833 
4.66905 
4.67974 

9.39393 
9.52813 
9.66360 

9.80034 
9.93838 
10.0777 

10.2183 
10.3602 
10.5035 

1.28261 
1.28463 
1.28665 

1.28866 
1.29066 
1.29266 

1.29465 
1.29664 

1.29862 

2.76330 
2.76766 
2.77200 

2.77633 
2.78065 
2.78495 

2.78924 
2.79352 
2.79779 

5.95334 
5.96273 
5.97209 

5.98142 
5.99073 
6.00000 

6.00925 
6.01846 
6.02765 

.473934 
.471698 
.469434 

.467290 
.465116 
.462963 

.460829 
.458716 
.456621 

2.20 

4.8400 

1.48324 

4.69042 

10.6480 

1.30059 

2.80204 

6.03681 

.454545 

2.21 
2.22 
2.23 

2.24 
2.25 
2.26 

2.27 

2.28 
2.29 

4.8841 
4.9284 
4.9729 

5.0176 
5.0625 
5.1076 

5.1529 
5.1984 
5.2441 

1.48661 
1.48997 
1.49332 

1.49666 
1.50000 
1.50333 

1.50665 
1.50997 
1.51327 

4.70106 
4.71169 
4.72229 

4.73286 
4.74342 
4.75395 

4.76445 
4.77493 
4.78539 

10.7939 
10.9410 
11.0896 

11.2394 
11.3906 
11.5432 

11.6971 
11.8524 
12.0090 

1.30256 
1.30452 
1.30648 

1.30843 
1.31037 
1.31231  . 

1.31424 
1.31617 
1.31809 

2.80628 
2.81050 

2.81472 

2.81892 
2.82311 
2.82728 

2.83145 
2.83560 

2.83974 

6.04594 
6.05505 
6.06413 

6.07318 

6.08220 
6.09120 

6.10017 
6.10911 
6.11803 

.452489 
.450450 
.448430 

.446429 
.444444 
.442478 

.440529 

.438596 
.436681 

2.30 

5.2900 

1.51658 

4.79583 

12.1670 

1.32001 

2.84387 

6.12693 

.434783 

2.31 
2.32 
2.33 

2.34 
2.35 
2.36 

2.37 

2.38 
2.39 

5.3361 
5.3824 
5.4289 

5.4756 
5.5225 
5.5696 

5.6169 
5.6644 
5.7121 

1.51987 
1.52315 
1.52643 

1.52971 
1.53297 
1.53623 

1.53948 
1.54272 
1.54596 

4.80625 
4.81664 
4.82701 

4.83735 
4.84768 
4.85798 

4.86826 
4.87852 
4.88876 

12.3264 
12.4872 
12.6493 

12.8129 
12.9779 
13.1443 

13.3121 
13.4813 
13.6519 

1.32192 
1.32382 
1.32572 

1.32761 
1.32950 
1.33139 

1.33326 
1.33514 
1.33700 

2.84798" 
2.85209 
2.856.18 

2.86026 
2.86433 
2.86838 

2.87243 
2.87646 
2.88049 

6.13579 
6.14463 
6.15345 

6.16224 
6.17101 
6.17975 

6.18846 
6.19715 

6.20582 

.432900 
.431034 
.429185 

.427350 
.425532 
.423729 

.421941 
.420168 
.418410 

2.40 

5.7600 

1.54919 

4.89898 

13.8240 

1.33887 

2.88450 

6.21447 

.416667 

2.41 
2.42 
2.43 

2.44 
2.45 
2.46 

2.47 
2.48 
2.49 

5.8081 
5.8564 
5.9049 

5.9536 
6.0025 
6.0516 

6.1009 
6.1504 
6.2001 

1.55242 
1.55563 
1.55885 

1.56205 
1.56525 
1.56844 

1.57162 
1.57480 
1.57797 

4.90918 
4.91935 
4.92950 

4.93964 
4.94975 
4.95984 

4.96991 
4.97996 
4.98999 

13.9975 
14.1725 
14.3489 

14.5268 
14.7061 
14.8869 

15.0692 
15.2530 
15.4382 

1.34072 
1.34257 
1.34442 

1.34626 
1.34810 
1.34993 

1.35176 
1.35358 
1.35540 

2.88850 
2.89249 
2.89647 

2.90044 
2.90439 
2.90834 

2.91227 
2.91620 
2.92011 

6.22308 
6.23168 
6.24025 

6.24880 
6.25732 
6.26583 

6.27431 
6.28276 
6.29119 

.414938 
.413223 
.411523 

.409836 
.408163 
.406504 

.404858 
.403226 
.401606 

2.50 

6.2500 

1.58114 

5.00000 

15.6250 

1.35721 

2.92402 

6.29961 

.400000 

n 

n^ 

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VI} 

Powers— Roots  — 

Reciprocals 

97 

n 

W2 

V^ 

vio^ 

n^ 

</n 

^10  w, 

1/n 

s/lOOn 

2.50 

6.2500 

1.58114 

6.00000 

15.6250 

1.35721 

2.92402 

6.29961 

.400000 

2.51 
2.52 
2.53 

2.54 
2.55 
2.56 

2.57 

2.58 
2.59 

6.3001 
6.3504 
6.4009 

6.4516 
6.5025 
6.5536 

6.6049 
6.6564 
6.7081 

1.58430 
1.58745 
1.59060 

1.59374 
1.59687 
1.60000 

1.60312 
1.60624 
1.60935 

5.00999 
5.01996 
5.02991 

5.03984 
5.04975 
5.05964 

5.06952 
5.07937 
5.08920 

15.8133 
16.0030 
16.1943 

16.3871 
16.5814 
16.7772 

16.9746 
17.1735 
17.3740 

1.35902 
1.36082 
1.36262 

1.36441 
1.36620 
1.36798 

1.36976 
1.37153 
1.37330 

2.92791 
2.93179 
2.93567 

2.93953 
2.94338 
2.94723 

2.95106 

2.95488 
2.95869 

6.30799 
6.31636 
6.32470 

6.33303 
6.34133 
6.34960 

6.35786 
6.36610 
6.37431 

.398406 
.396825 
.395257 

.393701 
.392157 
.390625 

.389105 
.387597 
.386100 

2.60 

6.7600 

1.61245 

5.09902 

17.5760 

1.37507 

2.96250 

6.38250 

.384(515 

2.61 
2.62 
2.63 

2.64 
2.65 
2.66 

2.67 
2.68 
2.69 

6.8121 
6.8644 
6.9169 

6.9696 
7.0225 
7.0756 

7.1289 
7.1824 
7.2361 

1.61555 
1.61864 
1.62173 

1.62481 

1.62788 
1.63095 

1.63401 
1.63707 
1.64012 

5.10882 
5.11859 
5.12835 

5.13809 
5.14782 
5.15752 

5.16720 
5.17687 
5.18652 

17.7796 

17.9847 
18.1914 

18.3997 
18.6096 
18.8211 

19.0342 
19.2488 
19.4651 

1.37683 
1.37859 
1.38034 

1.38208 
1.38383 
1.38557 

1.38730 
1.38903 
1.39076 

2.96629 
2.97007 
2.97385 

2.97761 
2.98137 
2.98511 

2.98885 
2.99257 
2.99629 

6.39068 
6.39883 
6.40696 

6.41507 
6.42316 
6.43123 

6.43928 
6.44731 
6.45531 

.383142 
.381679 
.380228 

-378788 
.377358 
.375940 

.374532 
.373134 
.371747 

2.70 

7.2900 

1.64317 

5.19615 

19.6830 

1.39248 

3.00000 

6.46330 

.370370 

2.71 

2.72 
2^73 

2.74 
2.75 
2.76 

2.77 
2.78 
2.79 

7.3441 
7.3984 
7.4529 

7.5076 
7.5625 
7.6176 

7.6729 

7.7284 
7.7841 

1.64621 
1.64924 
1.65227 

1.65529 
1.65831 
1.66132 

1.66433 
1.66733 
1.67033 

5.20577 
5.21536 
5.22494 

5.23450 
5.24404 
5.25357 

5.26308 
5.27257 
5.28205 

19.9025 
20.1236 
20.3464 

20.5708 
20.7969 
21.0246 

21.2539 
21.4850 
21.7176 

1.39419 
1.39591 
1.39761 

1.39932 
1.40102 
1.40272 

1.40441 
1.40610 
1.40778 

3.00370 
3.00739 
3.01107 

3.01474 
3.01841 
3.02206 

3.02570 
3.02934 
3.03297 

6.47127 
6.47922 
6.48715 

6.49507 
6.5029<j 
6.51083 

6.51868 
6.52652 
6.53434 

.369004 
.367647 
.366300 

.364964 
.363636 
.362319 

.361011 
.359712 
.358423 

2.80 

7.8400 

1.67332 

5.29150 

21.9520 

1.40946 

3.03659 

6.54213 

.357143 

2.81 
2.82 
2.83 

2.84 
2.85 
2.86 

2.87 
2.88 
2.89 

7.8961 
7.9524 
8.0089 

8.0656 
8.1225 
8.1796 

8.2369 
8.2944 
8.3521 

1.67631 
1.67929 
1.68226 

1.68523 
1.68819 
1.69115 

1.69411 
1.69706 
1.70000 

5.30094 
5.31037 
5.31977 

5.32917 
5.33854 
5.34790 

5.35724 
5.36656 
5.37587 

22.1880 
22.4258 
22.6652 

22.9063 
23.1491 
23.3937 

23.6399 
23.8879 
24.1376 

1.41114 
1.41281 
1.41448 

1.41614 
1.41780 
1.41946 

1.42111 
1.42276 
1.42440 

3.04020 
3.04380 
3.04740 

3.05098 
3.05456 
3.05813 

3.06169 
3.06524 
3.06878 

6.54991 

6.55767 
6.56541 

6.57314 

6.58084 
6.58853 

6.59620 
6.60385 
6.61149 

.355872 
.354610 
.353357 

.352113 

.350877 
.349650 

.348432 
.347222 
.346021 

2.90 

8.4100 

1.70294 

5.38516 

24.38fX) 

1.42604 

3.07232 

6.61911 

.344828 

2.91 
2.92 
2.93 

2.94 
2.95 
2.96 

2.97 

2.98 

'2.99 

8.4681 
8.5264 
8.5849 

8.6436 
8.7025 
8.7616 

8.8209 
8.8804 
8.9401 

1.70587 
1.70880 
1.71172 

1.71464 
1.71756 
1.72047 

1.72337 
1.72627 
1.72916 

5.39444 
5.40370 
5.41295 

5.42218 
5.43139 
5.44059 

5.44977 
5.45894 
5.46809 

24.6422 
24.8971 
25.1538 

25.4122 
25.6724 
25.9343 

26.1981 
26.46:56 
26.7309 

1.42768 
1.42931 
1.43094 

1.43257 
1.43419 
1.43581 

1.43743 
1.43904 
1.44065 

3.07584 
3.07936 
3.08287 

3.08638 
3.08987 
3.09336 

3.09684 
3.10031 
3.10378 

6.62671 
6.63429 
6.64185 

6.64940 
6.65693 
6.66444 

6.67194 
6.67942 
6.68688 

.343643 
.342466 
.341297 

.340136 
.338983 
.337838 

.336700 
.335570 
.334448 

3.00 

9.0000 

1.73205 

5.47723 

27.0000 

1.44225 

3.10723 

6.69433 

.333333 

n 

n2 

Vn 

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98 

Powers  —  Roots  — 

Reciprocals 

[VJ 

n 

n2 

Vn 

VIOm^ 

n^ 

^ 

^liin 

1/n 

^100  n 

3.00 

9.0000 

1.73205 

5.47723 

27.0000 

1.44225 

3.10723 

6.69433 

.333333 

3.01 
3.02 
3.03 

3.04 
3.05 
3.06 

3.07 
3.08 
3.09 

9.0601 
9.1204 
9.1809 

9.2416 
9.3025 
9.3636 

9.4249 
9.4864 
9.5481 

1.73494 
1.73781 
1.74069 

1.74356 
1.74642 
1.74929 

1.75214 
1.75499 
1.75784 

5.48635 
5.49545 
5.50454 

5.51362 
5.52268 
5.53173 

5.54076 
5.54977 

5.55878 

27.2709 
27.5436 
27.8181 

28.0945 
28.3726 
28.6526 

28.9344 
29.2181 
29.5036 

1.44385 
1.44545 
1.44704 

1.44863 
1.45022 
1.45180 

1.45338 
1.45496 
1.45653 

3.11068 
3.11412 
3.11756 

3.12098 
3.12440 
3.12781 

3.13121 
3.13461 
3.1.3800 

6.70176 
6.70917 
6.71657 

6.72395 
6.73132 
6.73866 

6.74600 
6.75331 
6.7(5061 

.332226 
.331126 
.330033 

.328947 
.327869 
.326797 

.325733 
.324675 
.323625 

3.10 

9.6100 

1.76068 

5.56776 

29.7910 

1.45810 

3.14138 

6.76790 

.322581 

3.11 
3.12 
3.13 

3.14 
3.15 
3.16 

3.17 
3.18 
3.19 

9.6721 
9.7344 
9.7969 

9.8596 
9.9225 
9.9856 

10.0489 
10.1124 
10.1761 

1.76352 
1.76635 
1.76918 

1.77200 
1.77482 
1.77764 

1.78045 
1.78326 
1.78606 

5.57674 
5.58570 
5.59464 

5.60357 
5.61249 
5.62139 

5.63028 
5.63915 
5.64801 

30.0802 
30.3713 
30.6643 

30.9591 
31.2559 
31.5545 

31.8550 
32.1574 
32.4618 

1.45967 
1.46123 
1.46279 

1.46434 
1.46590 
1.46745 

1.46899 
1.47054 

1.47208 

3.14475 
3.14812 
3.15148 

3.15483 
3.15818 
3.16152 

3.16485 
3.16817 
3.17149 

6.77517 
6.78242 
6.78966 

6.79(588 
6.80409 
6.81128 

6.81846 
6.82562 
6.83277 

.321543 
.320513 
.319489 

.318471 
.317460 
.316456 

.315457 
.314465 
.313480 

3.20 

10.2400 

1.78885 

5.65685 

32.7680 

1.47361 

3.17480 

6.83990 

.312500 

3.21 
3.22 
3.23 

3.24 
3.25 
3.26 

3.27 
3.28 
3.29 

10.3041 
10.3684 
10.4329 

10.4976 
10.5625 
10.6276 

10.6929 
10.7584 
10.8241 

1.79165 
1.79444 
1.79722 

1.80000 
1.80278 
1.80555 

1.80831 
1.81108 
1.81384 

5.()6569 
5.67450 
5.68331 

5.69210 

5.70088 
5.70964 

5.71839 
6.72713 

5.73585 

33.0762 
33.3862 
33.6983 

34.0122 
34.3281 
34.6460 

34.9658 
35.2876 
35.6113 

1.47515 
1.47668 
1.47820 

1.47973 
1.48125 

1.48277 

1.48428 
1.48579 
1.48730 

3.17811 
3.18140 
3.18469 

3.18798 
3.19125 
3.19452 

3.19778 
3.20104 
3.20429 

6.84702 
6.85412 
6.86121 

6.86829 
6.87534 
6.88239 

6.88942 
6.89643 
6.90344 

.311526 
.310559 
.309598 

.308642 
.307692 
.306748 

.305810 
.304878 
.303951 

3.30 

10.8900 

1.81659 

5.74456 

35.9370 

1.48881 

3.20753 

6.91042 

.303030 

3.31 
3.32 
3.33 

3.34 
3.35 
3.36 

3.37 

3.38 
3.39 

10.9561 
11.0224 
11.0889 

11.1556" 
11.2225 
11.2896 

11.3569 
11.4244 
11.4921 

1.81934 
1.82209 
1.82483 

1.82757 
1.83030 
1.83303 

1.83576 
1.83848 
1.84120 

5.75326 
5.76194 
5.77062 

5.77927 

5.78792 
5.79655 

5.80517 
5.81378 
5.82237 

36.2647 
36.5944 
36.9260 

37.2597 
37.5954 
37.9331 

38.2728 
38.6145 
38.9582 

1.49031 
1.49181 
1.49330 

1.49480 
1.49()29 
1.49777 

1.49926 
1.50074 
1.50222 

3.21077 
3.21400 
3.21722 

3.22044 
3.22365 
3.22686 

3.2.3006 
3.23325 
3.23643 

6.91740 
6.92436 
6.93130 

6.93823 
6.94515 
6.95205 

6.95894 
6.96582 
6.97268 

.302115 
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.300300 

.299401 
.298507 
.297619 

.296736 
.295858 
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3.40 

11.5600 

1.84391 

5.83095 

39.3040 

1.50369 

3.23961 

6.97953 

.294118 

3.41 
3.42 
3.43 

3.44 
3.45 
3.46 

3.47 

3.48 
3.49 

11.6281 
11.6964 
11.7649 

11.8336 
ll.^K)25 
11.9716 

12.0409 
12.1104 
12.1801 

1.84662 
1.84932 
1.85203 

1.85472 
1.85742 
1.86011 

1.86279 
1.86548 
1.86815 

5.83952 
5.84808 
5.85662 

5.86515 

5.87367 
5.88218 

5.89067 
5.89915 
5.90762 

39.6518 
40.0017 
40.3536 

40.7076 
41.0636 
41.4217 

41.7819 
42.1442 
42.5085 

1.50517 
1.50664 
1.50810 

1.50957 
1.51103 
1.51249 

1.51394 
1.51540 
1.51685 

3.24278 
3.24595 
3.24911 

3.25227 
3.25542 
3.25856 

3.26169 
3.26482 
3.26795 

6.98637 
6.99319 
7.00000 

7.00680 
7.01358 
7.02035 

7.02711 
7.03385 
7.04058 

.293255 
.292398 
.291545 

.290698 
.289855 
.289017 

.288184 
.287356 
.286533 

3.50 

12.2500 

1.87083 

5.91608 

42.8750 

1.51829 

3.27107 

7.04730 

.285714 

n 

n2 

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Powers  —  I 

loots  — 

Recipi 

'ocals 

99 

n 

n? 

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VlOn, 

n^ 

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1/n 

S/lOOn 

3.50 

12.2500 

1.87083 

5:91608 

42.8750 

1.51829 

3.27107 

7.04730 

.285714 

3.51 
3.52 
3.53 

3.54 
3.55 
3.56 

3.57 
3.58 
3.59 

12.3201 
12.3904 
12.4609 

12.5316 
12.6025 
12.6736 

12.7449 
12.8164 

12.8881 

1.87350 
1.87617 

1.87883 

1.88149 
1.88414 
1.88680 

1.88944 
1.89209 
1.89473 

5.92453 
5.93296 
5.94138 

5.94979 
5.95819 
5.96657 

5.97495 
5.98331 
5.99166 

43.2436 
43.6142 
43.9870 

44.3619 

44.7389 
45.1180 

45.4993 

45.8827 
46.2683 

1.51974 
1.52118 
1.52262 

1.52406 
1.52549 
1.52692 

1.52835 
1.52978 
1.53120 

3.27418 
3.27729 
3.28039 

3.28348 
3.28657 
3.28965 

3.29273 
3.29580 
3.29887 

7.05400 
7.06070 
7.06738 

7.07404 
7.08070 
7.08734 

7.09397 
7.10059 
7.10719 

.284900 
.284091 
.283286 

.282486 
.281690 
.280899 

.280112 
.279330 

.278552 

3.60 

12.9600 

1.89737 

6.00000 

46.6560 

1.53262 

3.30193 

7.11379 

.277778 

3.61 
3.62 
3.63 

3.64 
3.65 
3.66 

3.67 
3.68 
3.69 

13.0321 
13.1044 
13.1769 

13.2496 
13.3225 
13.3956 

13.4689 
13.5424 
13.6161 

1.90000 
1.90263 
1.90526 

1.90788 
1.91050 
1.91311 

1.91572 
1.91833 
1.92094 

6.00833 
6.01664 
6.02495 

6.03324 
6.04152 
6.04979 

6.05805 
6.06630 
6.07454 

47.0459 
47.4379 
47.8321 

48.2285 
48.6271 
49.0279 

49.4309 
49.8360 
50.2434 

1.53404 
1.53545 
1.53686 

1.53827 
1.53968 
1.54109 

1.54249 
1.54389 
1.54529 

3.30498 
3.30803 
3.31107 

3.31411 
3.31714 
3.32017 

3.32319 
3.32621 
3.32922 

7.12037 
7.12694 
7.13349 

7.14004 
7.14657 
7.15309 

7.15960 
7.16610 

7.17258 

.277008 
.276243 
.275482 

.274725 
.273973 
.273224 

.272480 
.271739 
.271003 

3.70 

13.6900 

1.92354 

6.08276 

50.6530 

1.54668 

3.33222 

7.17905 

.270270 

3.71 
3.72 
3.73 

3..74 
3.75 
3.76 

3.77 
3.78 
3.79 

13.7641 
13.8384 
13.9129 

13.9876 
14.0625 
14.1376 

14.2129 
14.2884 
14.3641 

1.92614 
1.92873 
1.93132 

1.93391 
1.93649 
1.93907 

1.94165 
1.94422 
1.94679 

6.09098 
6.09918 
6.10737 

6.11555 
6.12372 
6.13188 

6.14003 
6.14817 
6.15630 

51.0648 
51.4788 
51.8951 

52.3136 
52.7344 
53.1574 

53.5826 
54.0102 
54.4399 

1.54807 
1.54946 
1.55085 

1.55223 
1.55362 
1.55500 

1.55637 
1.55775 
1.55912 

3.33522 
3.33822 
3.34120 

3.34419 
3.34716 
3.35014 

3.35310 
3.35607 
3.35S)02 

7.18552 
7.19197 
7.19840 

7.20483 
7.21125 
7.21765 

7.22405 
7.23043 
7.23680 

.269542 
.268817 
.268097 

.267380 
.266667 
.265957 

.265252 
.264550 
.263852 

3.80 

14.4400 

1.94936 

6.16441 

54.8720 

1.56049 

3.36198 

7.24316 

.263158 

3.81 
3.82 
3.83 

3.84 
3.85 
3.86 

3.87 
3.88 
3.89 

14.5161 
14.5924 
14.6689 

14.7456 
14.8225 
14.8996 

14.9769 
15.0544 
15.1321 

1.95192 
1.95448 
1.95704 

1.95959 
1.96214 
1.96469 

1.96723 
1.96977 
1.97231 

6.17252 
6.18061 
6.18870 

6.19677 
6.20484 
6.21289 

6.22093 

6.22896 
6.23699 

55.3063 
55.7430 
56.1819 

56.6231 
57.0666 
57.5125 

57.9606 
58.4111 
58.8639 

1.56186 
1.56322 
1.56459 

1.56595 
1.56731 
1.56866 

1.57001 
1.57137 
1.57271 

3.36492 
3.36786 
3.37080 

3.37373 
3.37666 
3.37958 

3.38249 
3.38540 
3.38831 

7.24950 
7.25584 
7.26217 

7.26848 
7.27479 
7.28108 

7.28736 
7.293(i3 
7.29^)89 

.262467 
.261780 
.261097 

.260417 
.259740 
.259067 

.258398 
.257732 
.257069 

3.90 

15.2100 

1.97484 

6.24.500 

59.3190 

1.57406 

3.39121 

7.30614 

.256410 

3.91 
3.92 
3.93 

3.94 
3.95 
3.96 

3.97 
3.98 
3.99 

15.2881 
15.3664 
15.4449 

15.5236 
15.6025 
15.6816 

15.7609 
15.8404 
15.9201 

1.97737 
1.97990 
1.98242 

1.98494 
1.98746 
1.98997 

1.99249 
1.99499 
1.99750 

6.25300 
6.26099 
6.26897 

6.27694 
6.28490 
6.29285 

6.30079 
6.30872 
6.31664 

59.7765 
60.2363 
()0.6985 

61.1630 
61.6299 
62.0991 

62.5708 
63.0448 
63.5212 

1.57541 
1.57675 
1.57809 

1.57942 
1.58076 
1.58209 

1.58342 
1..58475 
1.58608 

3.39411 
3.39700 
3.39988 

3.40277 
3.40564 
3.40851 

3.41138 
3.41424 
3.41710 

7.31238 
7.31861 
7.32483 

7.33104 
7.33723 
7.34342 

7.34960 
7.35576 
7.36192 

.255754 
.255102 
.254453 

.253807 
.253165 
.252525 

.251889 
.251256 
.250627 

4.00 

16.0000 

2.00000 

6.32456 

64.0000 

1.58740 

3.41995 

7.36806 

.250000 

n 

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100 

Powers  — Roots  — 

Reciprocals 

[VI 

n 

n^ 

V^ 

VlOn 

n^ 

i^ 

</10n 

1/n 

</100n 

4.00 

16.0000 

2.00000 

6.32456 

64.0000 

1.58740 

3.41995 

7.36806 

.250000 

4.01 
4.02 
4.03 

4.04 
4.05 
4.06 

4.07 
4.08 
4.09 

16.0801 
16.1604 , 
16.2409 

16.3216 
16.4025 
16.4836 

16.5649 
16.6464 
16.7281 

2.00250 
2.00499 
2.00749 

2.00998 
2.01246 
2.01494 

2.01742 
2.01990 
2.02237 

6.33246 
6.34035 
6.34823 

6.35610 
6.36396 
6.37181 

6.37966 
6.38749 
6.39531 

64.4812 
64.9648 
65.4508 

65.9393 
66.4301 
66.9234 

67.4191 
67.9173 
68.4179 

1.58872 
1.59004 
1.59136 

1.59267 
1.59399 
1.59530 

1.59661 
1.59791 
1.59922 

3.42280 
3.42564 
3.42848 

3.43131 
3.43414 
3.43697 

3.43979 
3.44260 
3.44541 

7.37420 
7.38032 
7.38644 

7.39254 
7.39864 
7.40472 

7.41080 
7.41686 
7.42291 

.249377 
.248756 
.248139 

.247525 
.246914 
.246305 

.245700 
.245098 
.244499 

4.10 

16.8100 

2.02485 

6.40312 

68.9210 

1.60052 

3.44822 

7.42896 

.243902 

4.11 
4.12 
4.13 

4.14 
4.15 
4.16 

4.17 
4.18 
4.19 

16.8921 
16.9744 
17.0569 

17.1396 
17.2225 
17.3056 

17.3889 
17.4724 
17.5561 

2.02731 
2.02978 
2.03224 

2.03470 
2.03715 
2.03961 

2.04206 
2.04450 
2.04695 

6.41093 
6.41872 
6.42651 

6.43428 
6.44205 
6.44981 

6.45755 
6.46529 
6.47302 

69.4265 
69.9345 
70.4450 

70.9579 
71.4734 
71.9913 

72.5117 
73.0346 
73.6601 

1.60182 
1.60312 
1.60441 

1.60571 
1.60700 
1.60829 

1.60958 
1.61086 
1.61215 

8.45102 
3.45382 
3.45661 

3.45939 
3.46218 
3.46496 

3.46773 
3.47050 
3.47327 

7.43499 
7.44102 
7.44703 

7.45304 
7.45904 
7.46502 

7.47100 
7.47697 

7.48292 

.243309 
.242718 
.242131 

.241546 
.240964 
.240385 

.239808 
.239234 
.238663 

4.20 

17.6400 

2.04939 

6.48074 

74.0880 

1.61343 

3.47603 

7.48887 

.238095 

4.21 
4.22 
4.23 

4.24 
4.25 
4.26 

4.27 

4.28 
4.29 

17.7241 

17.8084 
17.8929 

17.9776 
18.0625 
18.1476 

18.2329 
18.3184 
18.4041 

2.05183 
2.05426 
2.05670 

2.05913 
2.06155 
2.06398 

2.06640 
2.06882 
2.07123 

6.48845 
6.49615 
6.50384 

6.51153 
6.51920 
6.52687 

6.53452 
6.54217 
6.54981 

74.6185 
75.1514 
75.6870 

76.2250 
76.7656 
77.3088 

77.8545 
78.4028 
78.9536 

1.61471 
1.61599 
1.61726 

1.61853 
1.61981 
1.62108 

1.62234 
1.62361 
1.62487 

3.47878 
3.48154 
3.48428 

3.48703 
3.48977 
3.49250 

3.49523 
3.49796 
3.50068 

7.49481 
7.50074 
7.50666 

7.51257 
7.51847 
7.52437 

7.53025 
7.53612 
7.54199 

.237530 
.236967 
.236407 

.235849 
.235294 
.234742 

.234192 
.233645 
.233100 

4.30 

18.4900 

2.07364 

6.55744 

79.5070 

1.62613 

3.50340 

7.54784 

.232558 

4.31 
4.32 
4.33 

4.34 
4.35 
4.36 

4.37 
4.38 
4.39 

18.5761 
18.6624 
18.7489 

18.8356 
18.9225 
19.0096 

19.0969 
19.1844 
19.2721 

2.07605 
2.07846 
2.08087 

2.08327 
2.08567 
2.08806 

2.09045 
2.09284 
2.09523 

6.56506 
6.57267 
6.58027 

6.58787 
6.59545 
6.60303 

6.61060 
6.61816 
6.62571 

80.0630 
80.6216 
81.1827 

81.7465 
82.3129 

82.8819 

83.4535 
84.0277 
84.6045 

1.62739 
1.62865 
1.62991 

1.63116 
1.63241 
1.63366 

1.63491 
1.63619 
1.63740 

3.50611 

3.50882 
3.51153 

3.51423 
3.51692 
3.51962 

3.52231 
3.52499 
3.52767 

7.55369 
7.55953 
7.56535 

7.57117 
7.57698 
7.58279 

7.58858 
7.59436 
7.60014 

.232019 
.231481 
.230947 

.230415 
.229885 
.229358 

.228833 
.228311 
.227790 

4.40 

19.3600 

2.09762 

6.63325 

85.1840 

1.63864 

3.53035 

7.60590 

.227273 

4.41 
4.42 
4.43 

4.44 
4.45 
4.46 

4.47 
4.48 
4.49 

19.4481 
19.5364 
19.6249 

19.7136 
19.8025 
19.8916 

19.9809 
20.0704 
20.1601 

2.10000 
2.10238 
2.10476 

2.10713 
2.10950 
2.11187 

2.11424 
2.11660 
2.11896 

6.64078 
6.64831 
6.65582 

6.66333 
6.67083 
6.67832 

6.68581 
6.69:528 
6.70075 

85.7661 
86.3509 
86.9383 

87.5284 
88.1211 
88.7165 

89.3146 
89.9154 
90.5188 

1.63988 
1.64112 
1.64236 

1.64359 
1.64483 
1.64606 

1.64729 
1.64851 
1.64974 

3.53302 
3.53569 
3.53835 

3.54101 
3.54367 
3.54632 

3.54897 
3.55162 
3.55426 

7.61166 
7.61741 
7.62315 

7.62888 
7.63461 
7.64032 

7.64603 
7.65172 
7.6.5741 

.226757 
.226244 
.225734 

.225225 
.224719 
.224215 

.223714 
.223214 
.222717 

4.60 

20.2500 

2.12132 

6.70820 

91.1250 

1.65096 

3.55689 

7.66.309 

.222222 

n 

n2 

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n^ 

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^10^ 

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102 

Powers  —  Boots  — 

Reciprocals 

[VI 

n 

n^ 

Vn 

VlOn 

n^ 

^ 

</10n 

1/n 

S/lOOn 

5.00 

25.0000 

2.23607 

7.07107 

125.000 

1.70998 

3.68403 

7.93701 

.200000 

5.01 
5.02 
5.03 

5.04 
5.05 
6.06 

6.07 
5.08 
5.09 

25.1001 
25.2004 
25.3009 

25.4016 
25.5025 
25.6036 

25.7049 
25.8064 
25.9081 

2.23830 
2.24054 
2.24277 

2.24499 
2.24722 
2.24944 

2.25167 
2.25389 
2.25610 

7.07814 
7.08520 
7.09225 

7.09930 
7.10634 
7.11337 

7.12039 
7.12741 
7.13442 

125.752 
126.506 
127.264 

128.024 

128.788 
129.554 

130.324 
131.097 
131.872 

1.71112 
1.71225 
1.71339 

1.71452 
1.71566 
1.71679 

1.71792 
1.7UX)5 
1.72017 

3  68649 
3.68894 
3.69138 

3.69383 
3.69627 
3.69871 

3.70114 
3.70357 
3.70600 

7.94229 
7.94757 
7.95285 

7.95811 
7.96337 
7.96863 

7.97387 
7.97911 
7.98434 

.199601 
.199203 
.198807 

.198413 
.198020 
.197628 

.197239 
.196850 
.196464 

5.10 

26.0100 

2.25832 

7.14143 

132.651 

1.72130 

3.70843 

7.98957 

.196078 

5.11 
5.12 
5.13 

5.14 
5.15 
5.16 

6.17 
5.18 
5.19 

26.1121 
26.2144 
26.3169 

26.4196 
26.5225 
26.6256 

26.7289 
26.8324 
26.9361 

2.26053 
2.2G274 
2.26495 

2.26716 
2.26936 
2.27156 

2.27376 
2.27596 
2.27816 

7.14843 
7.15542 
7.16240 

7.16938 
7.17635 
7.18331 

7.19027 
7.19722 
7.20417 

133.433 
134.218 
135.006 

135.797 
136.591 
137.388 

138.188 
138.992 
139.798 

1.72242 
1.72355 
1.72467 

1.72579 
1.72691 
1.72802 

1.72914 
1.73025 
1.73137 

3.71085 
3.71327 
3.71569 

3.71810 
3.72051 
3.72292 

3.72532 
372772 
3.73012 

7.99479 
8.00000 
8.00520 

8.01040 
8.01559 

8.02078 

8.02.596 
8.03113 
8.03629 

.195695 
.195312 
.194932 

.194553 
.194175 
.193798 

.193424 
.193050 
.192678 

5.20 

27.0400 

2.28035 

7.21110 

140.608 

1.73248 

3.73251 

8.04145 

.192308 

5.21 
5.22 
5.23 

5.24 
5.25 
5.26 

5.27 
5.28 
5.29 

27.1441 
27.2484 
27.3529 

27.4576 
27.5625 
27.6676 

27.7729 
27.8784 
27.9841 

2.28254 
2.28473 
2.28692 

2.28910 
2.29129 
2.29347 

2.29565 
2.29783 
2.30000 

7.21803 
7.22496 
7.23187 

7.23878 
7.24569 
7.25259 

7.25948 
7.26636 
7.27324 

141.421 
142.237 
143.056 

143.878 
144.703 
145.532 

146.363 
147.198 
148.036 

1.73359 
1.73470 
1.73580 

1.73691 
1.73801 
1.73912 

1.74022 
1.74132 
1.74242 

3.73490 
3.73729 
3.73968 

3.74206 
3.74443 
3.74681 

3.74918 
3.75155 
3.75392 

8.04660 
8.05175 
8.05689 

8.06202 
8.06714 
8.07226 

8.07737 
8.08248 
8.08758 

.191939 
.191571 
.191205 

.190840 
.190476 
.190114 

.189753 
.189394 
.189036 

5.30 

28.0900 

2.30217 

7.28011 

148.877 

1.74351 

3.75629 

8.09267 

.188679 

5.31 
5.32 
5.33 

6.34 
5.35 
5.36 

5.37 
5.38 
5.39 

28.1961 
28.3024 
28.4089 

28.5156 
28.6225 
28.7296 

28.8369 
28.9444 
29.0521 

2.30434: 
2.30651 
2.30868 

2.31084 
2.31301 
2.31517 

2.31733 
2.31948 
2.32164 

7.28697 
7.29383 
7.30068 

7.30753 
7.31437 
7.32120 

7.32803 
7.33485 
7.34166 

149.721 
150.569 
151.419 

152.273 
153.130 
153.991 

154.854 
155.721 
156.591 

1.74461 
1.74570 
1.74680 

1.74789 

1.74898 
1.75007 

1.75116 
1.75224 
1.75333 

3.75865 
3.76101 
3.76336 

3.76571 
3.76806 
3.77041 

3.77276 
3.77509 
3.77743 

8.09776 
8.10284 
8.10791 

8.11298 
8.11804 
8.12310 

8.12814 
8.13319 
8.13822 

.188324 
.187970 
.187617 

.187266 
.186916 
.186567 

.186220 
.185874 
.185529 

5.40 

29.1600 

2.32379 

7.34847 

157.464 

1.75441 

3.77976 

8.14325 

.185185 

5.41 
5.42 
6.43 

6.44 
5.45 
6:46 

5.47 
5.48 
5.49 

29.2681 
29.3764 
29.4849 

29.5936 
29.7025 
29.8116 

29.9209 
30.0304 
30.1401 

2.32594 
2.32809 
2.33024 

2.33238 
2.33452 
2.33666 

2.33880 
2.34094 
2.34307 

7.35527 
7.36206 
7.36885 

7.37564 
7.38241 
7.38918 

7.39594 
7.40270 
7.40945 

158.340 
159.220 
160.103 

160.989 
161.879 
162.771 

163.667 
164.567 
165.469 

1.75549 
1.75657 
1.75765 

1.75873 
1.75981 
1.76088 

1.76196 
1.76303 
1.76410 

3.78209 
3.78442 
3.78675 

3.78907 
3.79139 
3.79371 

3.79603 
3.79834 
3.80065 

8.14828 
8.15329 
8.15831 

8.16331 
8.16831 
8.17330 

8.17829 
8.18327 
8.18824 

.184843 
.184502 
.184162 

.183824 
.183486 
.183150 

.182815 
.182482 
.182149 

5.50 

30.2500 

2.34521 

7.41620 

166.375 

1.76517 

3.80295 

8.19321 

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n 

n2 

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Powers  —  Roots  — 

Reciprocals 

103 

n 

n^ 

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5.50 

30.2500 

2.34521 

7.41620 

166.375 

1.76517 

3.80295 

8.19321 

.181818 

5.51 
5.52 
5.53 

5.54 
5.55 
5.56 

5.57 
5.58 
5.59 

30.3601 
30.4704 
30.5809 

30.6916 
30.8025 
30.9136 

31.0249 
31.1364 
31.2481 

2.34734 
2.34947 
2.35160 

2.35372 
2.35584 
2.35797 

2.36008 
2.36220 
2.36432 

7.42294 
7.42967 
7.43640 

7.44312 
7.44983 
7.45654 

7.46324 
7.46994 
7.47663 

167.284 
168.197 
169.112 

170.031 
170.954 
171.880 

172.809 
173.741 
174.677 

1.76624 
1.76731 
1.76838 

1.76944 
1.77051 
1.77157 

1.77263 
1.77369 
1.77475 

3.80526 
3.80756 
3.80985 

3.81215 
3.81444 
3.81673 

3.81^)02 
3.82130 

3.82358 

8.19818 
8.20313 
8.20808 

8.21303 
8.21797 
8.22290 

8.22783 
8.2.S275 
8.23766 

.181488 
.181159 
.180832 

.180505 
.180180 
.179856 

.179533 
.179211 
.178891 

5.60 

31.3600 

2.36643 

7.48331 

175.616 

1.77581 

3.8258(5 

8.24257 

.178571 

5.61 
5.62 
5.63 

5.64 
5.65 
5.66 

5.67 
5.68 
5.69 

31.4721 
31.5844 
31.6969 

31.8096 
31.9225 
32.0356 

32.1489 
32.2624 
32.3761 

2.36854 
2.37065 
2.37276 

2.37487 
2.37697 
2.37908 

2.38118 

2.38328 
2.38537 

7.48999 
7.49(567 
7.50333 

7.50999 
7.51665 
7.52330 

7.52994 
7.53658 
7.54321 

176.558 
177.504 
178.454 

179.406 
180.362 
181.321 

182.284 
183.250 
184.220 

1.77686 
1.77792 
1,77897 

1.78003 
1.78108 
1.78213 

1.78318 
1.78422 
1.78527 

3.82814 
3.83041 
3.83268 

3.83495 
3.83722 
3.83948 

3.84174 
3.84399 
3.84625 

8.24747 
8.25237 
8.25726 

8.26215 
8.26703 
8.27190 

8.27677 
8.28164 
8.28649 

.178253 
.177936 
.177620 

.177305 
.176991 
.176678 

.176367 
.176056 
.175747 

5.70 

32.4900 

2.38747 

7.54983 

185.193 

1.78632 

3.84850 

8.29134 

.175439 

.0.71 

5.72 
5.73 

5.74 
5.75 
5.76 

5.77 

5.78 
5.79 

32.6041 
32.7184 
32.8329 

32.9476 
33.0625 
33.1776 

33.2929 
33.4084 
33.5241 

2.38956 
2.39165 
2.39374 

2.39583 
2.39792 
2.40000 

2.40208 
2.40416 
2.40624 

7.55645 
7.56307 
7.56968 

7.57628 
7.58288 
7.58947 

7.59605 
7.60263 
7.60920 

186.169 
187.149 
188.133 

189.119 
190.109 
191.103 

192.100 
193.101 
194.105 

1.78736 
1.78840 
1.78944 

1.79048 
1.79152 
1.79256 

1.79360 
1.79463 
1.79567 

3.85075 
3.85300 
3.85524 

3.85748 
3.85972 
3.86196 

3.86419 
3.86642 
3.86865 

8.29619 
8.30103 
8.30587 

8.31069 
8.31552 
8.32034 

8.32515 
8.32995 
8.33476 

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.174216 
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5.80 

33.6400 

2.40832 

7.61577 

195.112 

1.79670 

3.87088 

8.33955 

.172414 

5.81 
5.82 
5.83 

5.84 
5.85 
5.86 

5.87 
5.88 
5.89 

33.7561 
33.8724 
33.9889 

34.1056 
34.2225 
34.3396 

34.4569 
34.5744 
34.6921 

2.41039 
2.41247 
2.41454 

2.41661 
2.41868 
2.42074 

2.42281 

2.42487 
2.42693 

7.62234 

7.62889 
7.63544 

7.64199 
7.64853 
7.65506 

7.66159 
7.66812 
7.67463 

196.123 
197.137 
198.155 

199.177 
200.202 
201.230 

202.262 
203.297 
204.336 

1.79773 
1.79876 
1.79979 

1.80082 
1.80185 
1.80288 

1.80390 
1.80492 
1.80595 

3.87310 
3.87532 
3.87754 

3.87975 
3.88197 
3.88418 

3.88639 
3.88859 
3.89080 

8.34434 
8.34913 
8.35390 

8.35868 
8.36345 
8.36821 

8.37297 
8.37772 
8.38247 

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5.90 

34.8100 

2.42899 

7.68115 

205.379 

1.80697 

3.89300 

8.38721 

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5.91 
5.92 
5.93 

5.94 
5.95 
5.96 

5.97 
5.98 
5.99 

34.9281 
35.0464 
35.1649 

35.2836 
35.4025 
35.5216 

35.6409 
35.7604 
35.8801 

2.43105 
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2.43516 

2.43721 
2.43926 
2.44131 

2.44336 
2.44540 
2.44745 

7.68765 
7.69415 
7.70065 

7.70714 
7.71362 
7.72010 

7.72658 
7.73305 
7.73951 

206.425 
207.475 
208.528 

209.585 
210.645 
211.709 

212.776 
213.847 
214.922 

1.80799 
1.80901 
1.81003 

1.81104 
1.81206 
1.81307 

1.81409 
1.81510 
1.81611 

3.89519 
3.89739 
3.89958 

3.90177 
3.90396 
3.90615 

3.90833 
3.91051 
3.91269 

8.39194 
8.39667 
8.40140 

8.40612 
8  41083 
8.41554 

8.42025 
8.42494 
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36.0000 

2.44949 

7.74597 

216.000 

1.81712 

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2.44949 

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3.91487 

8.43433 

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6.01 
6.02 
6.03 

6.04 
6.05 
6.06 

6.07 
6.08 
6.09 

36.1201 
36.2404 
36.3609 

36.4816 
36.6025 
36.7236 

36.8449 
36.9664 
37.0881 

2.45153 
2.45357 
2.45561 

2.45764 
2.45967 
2.46171 

2.46374 
2.46577 
2.46779 

7.75242 
7.75887 
7.76531 

7.77817 
7.78460 

7.79102 
7.79744 

7.80385 

217.082 
218.167 
219.256 

220.349 
221.445 
222.545 

223.649 
224.756 

225.867 

1.81813 
1.81914 
1.82014 

1.82115 
1.82215 
1.82316 

1.82416 
1.82516 
1.82616 

3.91704 
3.91921 
3.92138 

3.92355 
3.92571 
3.92787 

3.93003 
3.93219 
3.93434 

8.43901 
8.44369 
8.44836 

8.45303 
8.45769 
8.46235 

8.46700 
8.47165 
8.47629 

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6.10 

37.2100 

2.46982 

7.81025 

226.981 

1.82716 

3.93650 

8.48093 

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6.11 
6.12 
6.13 

6.14 
6.15 
6.16 

6.17 
6.18 
6.19 

37.3321 
37.4544 
37.5769 

37.69c^6 
37.8225 
37.9456 

38.0689 
38.1924 
38.3161 

2.47184 
2.47386 
2.47588 

2.47790 
2.47992 
2.48193 

2.48395 
2.48596 

2.48797 

7.81665 
7.82304 
7.82943 

7.83582 
7.84219 

7.84857 

7.85493 
7.86130 
7.86766 

228.099 
229.221 
230.346 

231.476 

232.608 
233.745 

234.885 
236.029 
237.177 

1.82816 
1.82915 
1.83015 

1.83115 
1.83214 
1.83313 

1.83412 
1.83511 
1.83610 

3.93865 
3.94079 
3.94294 

3.94508 
3.94722 
3.94936 

3.95150 
3.95363 
3.95576 

8.48556 
8.49018 
8.49481 

8.49942 
8.50403 
8.50864 

8.51324 

8.51784 
8.52243 

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.162866 
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6.20 

38.4400 

2.48998 

7.87401 

238.328 

1.83709 

3.95789 

8.52702 

.161290 

6.21 
6.22 
6.23 

6.24 
6.25 
6.26 

6.27 
6.28 
6.29 

38.5641 
38.6884 
38.8129 

38.9376 
39.0625 
39.1876 

39.3129 
39.4384 
39.5641 

2.49199 
2.49399 
2.49600 

2.49800 
2.50000 
2.50200 

2.50400 
2.50599 
2.50799 

7.88036 
7.88670 
7.89303 

7.89937 
7.90569 
7.91202 

7.91833 
7.92465 
7.93095 

239.483 
240.642 
241.804 

242.971 
244.141 
245.314 

246.492 
247.673 

248.858 

1.83808 
1.83906 
1.84005 

1.84103 
1.84202 
1.84300 

1.84398 
1.84496 
1.84594 

3.96002 
3.96214 
3.96427 

3.96638 
3.96850 
3.97062 

3.97273 

3.97484 
3.97695 

8.53160 
8.53618 
8.54075 

8.54532 
8.54988 
8.55444 

8.55899 
8.56354 
8.56808 

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6.30 

39.6900 

2.50998 

7.93725 

250.047 

1.84691 

3.97906 

8.57262 

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6.31 
6.32 
6.33 

6.34 
6.35 
6.36 

6.37 
6.38 
6,39 

39.8161 
39.9424 
40.0689 

40.1956 
40.3225 
40.4496 

40.5769 
40.7044 
40.8321 

2.51197 
2.51396 
2.51595 

2.51794 
2.51992 
2.52190 

2.52389 
2.52587 

2.52784 

7.94355 
7.94984 
7.95613 

7.96241 

7.96869 
7.97496 

7.98123 
7.98749 
7.99375 

251.240 
252.436 
253.636 

254.840 
256.048 
257.259 

258.475 
259.694 
260.917 

1.84789 
1.84887 
1.84984 

1.85082 
1.85179 
1.85276 

1.85373 
1.85470 

1.85567 

3.98116 
3.98326 
3.98536 

3.98746 
3.98956 
3.99165 

3.99374 
3.99583 
3.99792 

8.57715 
8.58168 
8.58620 

8.59072 
8.59524 
8.59975 

8.60425 
8.60875 
8.61325 

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6.40 

40.9600 

2.52982 

8.00000 

262.144 

1.85664 

4.00000 

8.61774 

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6.41 
6.42 
6.43 

6.44 
6.45 
6.46 

6.47 
6.48 
6.49 

41.0881 
41.2164 
41.3449 

41.4736 
41.6025 
41.7316 

41.8609 
41.9904 
42.1201 

2.53180 
2.53377 
2.53574 

2.53772 
2.53969 
2.54165 

2.54362 
2.54558 
2.54755 

8.00625 
8.01249 
8.01873 

8.02496 
8.03119 
8.03741 

8.04363 
8.04984 
8.05605 

263.375 
264.609 

265.848 

267.090 
268.336 
269.586 

270.840 
272.098 
273.359 

1.85760 
1.85857 
1.85953 

1.86050 
1.86146 
1.86242 

1.86338 
1.86434 
1.86530 

4.00208 
4.00416 
4.00624 

4.00832 
4.01039 
4.01246 

4.01453 
4.01660 
4.01866 

8.62222 
8.62671 
8.63118 

8.63566 
8.64012 
8.64459 

8.64904 
8.65350 
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2.54951 

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42.3801 
42.5104 
42.6409 

42.7716 
42.9025 
43.0336 

43.1649 
43.2904 
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2.55147 
2.55343 
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2.56320 
2.50515 
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8.06846 
8.07405 
8.08084 

8.08703 
8.09321 
8.09938 

8.10555 
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275.894 
277.168 
278.445 

279.726 
281.011 
282.300 

283.593 
284.890 
286.191 

1.86721 
1.80817 
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1.87008 
1.87103 
1.87198 

1.87293 

1.87388 
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4.02279 
4.02485 
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4.02896 
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4.03306 

4.03511 
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8.60083 
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44.0896 
44.2225 
44.3556 

44.4889 
44.0224 
44.7501 

2.57099 
2.57294 

2.57488 

2.57682 

2.57876 
2.58070 

2.58203 
2.58457 
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8.13634 
8.14248 

8.14862 
8.15475 
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8.16701 
8.17313 
8.17924 

288.805 
290.118 
291.434 

292.755 
294.080 
295.408 

296.741 
298.078 
299.418 

1.87072 

1.87707 
1.87862 

1.87956 

1.88050 

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1.88239 
1.88333 
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4.04.328 
4.04532 
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4.04939 
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4.05750 
4.05953 

8.71098 
8.71537 
8.71976 

8.72414 
8.72852 
8.73289 

8.73726 
8.74162 
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6.70 

44.8900 

2.58844 

8.18535 

300.763 

1.88520 

4.06155 

8.75034 

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0.71 

0.72 
0.73 

0.74 
0.75 
0.76 

0.77 
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45.0241 
45.1584 
45.2929 

45.4276 
45.5625 
45.6976 

45.8329 
45.9684 
40.1041 

2.59037 
2.59230 
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2.00192 
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8.19146 
8.19756 
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8.20975 
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8.22800 
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302.112 
303.464 
304.821 

306.182 
307.547 
308.916 

310.289 
311.066 
313.047 

1.88014 
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8.75409 
8.75904 
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8.70772 
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314.432 

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2.62107 
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8.25227 
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8.27043 
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315.821 
317.215 
318.612 

320.014 
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322.829 

324.243 
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47.7481 
47.8804 
48.0249 

48.1636 
48.3025 
48.4416 

48.5809 
48.7204 
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2.62869 
2.63059 
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2.64008 
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8.31204 
8.31805 
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329.939 
331.374 
332.813 

334.255 
335.702 
337.154 

338.609 
340.068 
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.142857 

7.01 
7.02 
7.03 

7.04 
7.05 
7.06 

7.07 
7.08 
7.09 

49.1401 
49.2804 
49.4209 

49.5616 
49.7025 
49.8436 

49.9849 
50.1264 
50.2681 

2.64764 
2.64953 
2.65141 

2.65330 
2.65518 
2.65707 

2.65895 
2.66083 
2.66271 

8.37257 
8.37854 
8.38451 

8.39047 
8.39643 
8.40238 

8.40833 
8.41427 
8.42021 

:344.472 
345.948 
347.429 

348.914 
350.403 
351.896 

353.393 
354.895 
3.56.401 

1.91384 
1.91475 
1.91566 

1.91657 
1.91747 
1.91838 

1.91929 
1.92019 
1.92109 

4.12325 
4.12521 
4.12716 

4.12912 
4.13107 
4.13303 

4.13498 
4.13693 

4.13887 

8.88327 
8.88749 
8.89171 

8.89592 
8.90013 
8.90434 

8.90854 
8.91274 
8.91693 

.142653 
.142450 
.142248 

.142045 
.141844 
.141643 

.141443 
.141243 
.141044 

7.10 

50.4100 

2.66458 

8.42615 

357.911 

1.92200 

4.14082 

8.92112 

.140845 

7.11 
7.12 
7.13 

7.14 
7.15 
7.16 

7.17 

7.18 
7.19 

50.5521 
50.6944 
50.8369 

50.9796 
51.1225 
51.2656 

51.4089 
51.5524 
51.6961 

2.66646 
2.66833 
2.67021 

2.67208 
2.67395 
2.67582 

2.67769 
2.67955 
2.68142 

8.43208 
8.43801 
8.44393 

8.44985 
8.45577 
8.46168 

8.46759 
8.47349 
8.47939 

359.425 
360.944 
362.467 

363.994 
365.526 
367.062 

368.602 
370.146 
371.695 

1.92290 
1.92380 
1.92470 

1.92560 
1.92650 
1.92740 

1.92829 
1.92919 
1.93008 

4.14276 
4.14470 
4.14664 

4.14858 
4.15052 
4.15245 

4.15438 
4.15631 
4.1.5824 

8.92531 
8.92949 
8.933(i7 

8.93784 
8.94201 
8.94618 

8.95034 
8.95450 
8.95866 

.140647 
.140449 
.140252 

.140056 
.139860 
.139665 

.139470 
.139276 
.139082 

7.20 

51.8400 

2.68328 

8.48528 

373.248 

1.93098 

4.16017 

8.96281 

.138889 

7.21 
7.22 
7.23 

7.24 
7.25 
7.26 

7.27 
7.28 
7.29 

51.9841 
52.1284 
52.2729 

52.4176 
52.5625 
52.7076 

52.8529 
52.9984 
53.1441 

2.68514 
2.68701 
2.68887 

2.69072 
2.69258 
2.69444 

2.69629 
2.69815 
2.70000 

8.49117 
8.49706 
8.50294 

8.50882 
8.51469 
8.52056 

8.52643 
8.53229 
8.53815 

374.805 
376.367 
377.933 

379.503 

381.078 
382.657 

384.241 

385.828 
387.420 

1.93187 
1.93277 
1.93366 

1.93455 
1.93544 
1.93633 

1.93722 
1.93810 
1.93899 

4.16209 
4.16402 
4.16594 

4.16786 
4.16978 
4.17169 

4.17361 
4.17552 
4.17743 

8.96696 
8.97110 
8.97524 

8.97938 
8.98351 
8.98764 

8.99176 
8.99588 
900000 

.138696 
.138504 
.138313 

.138122 
.137931 
.137741 

.137552 
.137363 
.137174 

7.30 

53.2900 

2.70185 

8.54400 

389.017 

1.93988 

4.17934 

9.00411 

.136986 

7.31 
7.32 
7.33 

7.34 
7.35 
7.36 

7.37 
7.38 
7.39 

53.4361 
53.5824 
53.7289 

53.8756 
54.0225 
54.1696 

54.3169 
54.4644 
54.6121 

2.70370 
2.70555 
2.70740 

2.70924 
2.71109 
2.71293 

2.71477 
2.71662 

2.71846 

8.54985 
8.55570 
8.56154 

8..56738 
8.57321 
8.57904 

8.58487 
8.59069 
8.59651 

390.618 
392.223 
393.833 

395.447 
397.065 
398.688 

400.316 
401.947 
403.583 

1.94076 
1.94165 
1.94253 

1.94341 
1.94430 
1.94518 

1.94606 
1.94()94 
1.94782 

4.18125 
4.18315 
4.18506 

4.18696 
4.18886 
4.19076 

4.19266 
4.19455 
4.19644 

9.00822 
9.01233 
9.01643 

9.02053 
9.02462 
9.02871 

9.03280 
9.03689 
9.04097 

.136799 
.136612 
.136426 

.136240 
.136054 
.135870 

.135685 
.135501 
.135318 

7.40 

54.7600 

2.72029 

8.60233 

405.224 

1.94870 

4.19834 

9.04504 

.135135 

7.41 
7.42 
7.43 

7.44 
7.45 
7.46 

7.47 

7.48* 

7.49 

54.9081 
55.0564 
55.2049 

55.3536 
55.5025 
55.6516 

55.8009 
55.9504 
56.1001 

2.72213 
2.72397 
2.72580 

2.72764 
2.72947 
2.73130 

2.73313 
2.73496 
2.73679 

8.60814 
8.61394 
8.61974 

8.62554 
8.63134 
8.63713 

8.64292 
8.64870 
8.65448 

406.869 
408.518 
410.172 

411.831 
413.494 
415.161 

416.833 
418.509 
420.190 

1.94957 
1.95045 
1.95132 

1.95220 
1.95307 
1.95.395 

1.95482 
1.95569 
1.95656 

4.20023 
4.20212 
4.20400 

4.20589 
4.20777 
4.20965 

4.21153 
4.21341 
4.21529 

9.04911 
9.05318 
9.05725 

9.06131 
9.06537 
9.06942 

9.07347 
9.07752 
9.08156 

.134953 
.134771 
.134590 

.134409 
.134228 
.134048 

.133869 
.133690 
.133511 

7.50 

56.2500 

2.73861 

8.66025 

421.875 

1.95743 

4.21716 

9.08560 

.133333 

n 

n^ 

Vn 

VlOn 

n^ 

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l/n 

^10^ 

^\mn 

VI] 

Powers  —  Roots  — 

Reciprocals 

107 

n 

VlOw, 

n^ 

^n 

^\^n 

\ln 

n? 

Vn 

^\mn 

7.60 

56.2500 

2.73861 

8.66025 

421.875 

1.95743 

4.21716 

9.08560 

.133333 

7.51 
7.52 
7.53 

7.54 
7.55 
7.56 

7.57 
7.58 
7.59 

56.4001 
56.5504 
56.7009 

56.8516 
57.0025 
57.1536 

57.3049 
57.4564 
57.6081 

2.74044 
2.74226 
2.74408 

2.74591 
2.74773 
2.74955 

2.75136 
2.75318 
2.75500 

8.66603 
8.67179 
8.67756 

8.68332 
8.68907 
8.69483 

8.70057 
8.70632 
8.71206 

423.565 
425.259 
426.958 

428.661 
430.369 
432.081 

433.798 
435.520 
437.245 

1.95830 
1.95917 
1.96004 

1.96091 
1.96177 
1.96264 

1.96350 
1.96437 
l.f)6523 

4.21904 
4  22091 
4.22278 

4.22465 
4.22651 
4.22838 

4.23024 
4.23210 
4.23396 

9.08964 
9.09367 
9.09770 

9.10173 
9.10575 
9.10977 

9.11378 
9.11779 
9.12180 

.133156 
.132979 
.132802 

.132626 
.132450 
.132275 

.132100 
.131926 
.131752 

7.60 

57.7600 

2.75681 

8.71780 

438.976 

1.96610 

4.23582 

9.12581 

.131579 

7.61 
7.62 
7.63 

7.64 
7.65 
7.66 

7.67 
7.68 
7.69 

57.9121 
58.0644 
58.2169 

58.3696 
58.5225 
58.6756 

58.8289 
58.9824 
59.1361 

2.75862 
2.76043 
2.76225 

2.76405 
2.76586 
2.76767 

2.76948 
2.77128 
2.77308 

8.72353 
8.7292() 
8.73499 

8.74071 
8.74643 
8.75214 

8.75785 
8.76356 
8.76926 

440.711 
442.451 
444.195 

445.944 
447.697 
449.455 

451.218 
452.985 
454.757 

1.96696 
1.96782 
1.96868 

1.96954 
1.97040 
1.97126 

1.97211 
1.97297 
1.97383 

4.23768 
4.23954 
4.24139 

4.24324 
4.24509 
4.24694 

4.24879 
4.25063 
4.25248 

9.12981 
9.13380 
9.13780 

9.14179 
9.14577 
9.14976 

9.15374 
9.15771 
9.16169 

.131406 
.131234 
.131062 

.130890 
.130719 
.130548 

.130378 
.130208 
.130039 

7.70 

59.2900 

2.77489 

8.77496 

456.533 

1.97468 

4.25432 

9.16566 

.129870 

7.71 

7.72 
7.73 

7.74 

7.75 
7.76 

7.77 
7.78 
7.79 

59.4441 
59.5984 
59.7529 

59.9076 
60.0625 
60.2176 

60.3729 
60.5284 
60.6841 

2.77669 
2.77849 
2.78029 

2.78209 
2.78388 
2.78568 

2.78747 
2.78927 
2.79106 

8.78066 
8.78635 
8.79204 

8.79773 
8.80341 
8.80909 

8.81476 
8.82043 
8.82610 

458.314 
460.100 
461.8i:)0 

463.685 
465.484 
467.289 

469.097 
470.911 
472.729 

1.97554 
1.97639 
1.97724 

1.97809 
1.97895 
1.97980 

1.98065 
1.98150 
1.98234 

4.25616 
4.25800 
4.25984 

4.26167 
4.26351 
4.26534 

4.26717 
4.26900 
4.27083 

9.16962 
9.17359 
9.17754 

9.18150 
9.18545 
9.18940 

9.19335 
9.19729 
9.20123 

.129702 
.129534 
.129366 

.129199 
.129032 
.128866 

.128700 
.128535 
.128370 

7.80 

7.81 

7.82 
7.83 

7.84 
7.85 
7.86 

7.87 
7.88 
7.89 

60.8400 

2.79285 

8.83176 

474.552 

1.98319 

4.27266 

9.20516 

.128205 

60.9961 
61.1524 
61.3089 

61.4656 
61.6225 
61.7796 

61.9369 
62.0944 
62.2521 

2.79464 
2.79643 

2.79821 

2.80000 
2.80179 
2.80357 

2.80535 
2.80713 
2.80891 

8.83742 
8.84308 
8.84873 

8.85438 
8.8f>002 
8.86566 

8.87130 
8.87694 
8.88257 

476.380 
478.212 
480.049 

481.890 
483.737 
485.588 

487.443 

489.304 
491.169 

1.98404 
1.98489 
1.98573 

1.98658 
1.98742 
1.98826 

1.98911 
1.98995 
1.99079 

4.27448 
4.27631 
4.27813 

4.27995 
4.28177 
4.28359 

4.28540 

4.28722 
4.28903 

9.20910 
9.21302 
9.21695 

9.22087 
9.22479 
9.22871 

9.23262 
9.23653 
9.24043 

.128041 
.127877 
.127714 

.127551 
.127389 
.127226 

.127065 
.126904 
.126743 

7.90 

62.4100 

2.81069 

8.88819 

493.039 

1.99163 

4.29084 

9.24434 

.126582 

7.91 
7.92 
7.93 

7.94 
7.95 
7.96 

7.97 
7.98 
7.99 

62.5681 
62.7264 
62.8849 

63.0436 
63.2025 
63.3616 

63.5209 
63.6804 
63.8401 

2.81247 
2.81425 
2.81603 

2.81780 
2.81957 
2.82135 

2.82.312 
2.82489 
2.82666 

8.89382 
8.89944 
8.90505 

8.91067 
8.91628 
8.92188 

8.92749 
8.93308 
8.93868 

494.914 
496.793 
498.677 

500.566 
502.460 
504.358 

506.262 
508.170 
510.082 

1.99247 
1.99331 
1.99415 

1.99499 
1.99582 
1.99666 

1.99750 
1.99833 
1.99917 

4.29265 
4.29446 
4.29627 

4.29807 
4.29987 
4.30168 

4.30348 
4.30528 
4.30707 

9.24823 
9.25213 
9.25602 

9.25991 
9.26380 
9.26768 

9.27156 
9.27544 
9.27931 

.126422 
.126263 
.126103 

.125945 
.125786 
.125628 

.125471 
.125313 
.125156 

8.00 

64.0000 

2.82843 

8.94427 

512.000 

2.00000 

4.30887 

9.28318 

.125000 

n 

1*2 

y/n 

n^ 

^n 

^\^n 

\ln 

VlOn 

^100  n 

108 

Powers  — Roots  — 

Reciprocals 

[VI 

n 

n^ 

Vn 

VlOn 

n^ 

^ 

^10  n 

l/n 

^100  n 

8.00 

64.0000 

2.82843 

8.94427 

512.000 

2.00000 

4.30887 

9.28318 

.125000 

8.01 
8.02 
8.03 

8.04 
8.05 
8.06 

8.07 
8.08 
8.09 

64.1601 
64.3204 
64.4809 

64.6416 
64.8025 
64.9636 

65.1249 
65.2864 
65.4481 

2.83019 
2.83196 
2.83373 

2.83549 
2.83725 
2.83901 

2.84077 
2.84253 
2.84429 

8.94986 
8.95545 
8.96103 

8.96660 
8.97218 
8.97775 

8.98.332 
8.98888 
8.99444 

513.922 
515.850 
517.782 

519.718 
521.660 
523.607 

525.558 
527.514 
529.475 

2.00083 
2.00167 
2.00250 

2.00333 
2.00416 
2.00499 

2.00582 
2.00664 
2.00747 

4.31066 
4.31246 
4.31425 

4.31604 
4.31783 
4.31961 

4.32140 
4.32318 
4.32497 

9.28704 
9.29091 
9.29477 

9.29862 
9.30248 
9.30633 

9.31018 
9.31402 
9.31786 

.124844 
.124688 
.124533 

.124378 
.124224 
.124069 

.123916 
.123762 
.123609 

8.10 

65.6100 

2.84605 

9.00000 

531.441 

2.00830 

4.32675 

9.32170 

.123457 

8.11 
8.12 
8.13 

8.14 
8.15 
8.16 

8.17 
8.18 
8.19 

65.7721 
65.9344 
66.0969 

66.2596 
66.4225 
66.5856 

66.7489 
66.9124 
67.0761 

2.84781 
2.84956 
2.85132 

2.85307 
2.85482 
2.85657 

2.85832 
2.86007 
2.86182 

9.00555 
9.01110 
9.01665 

9.02219 
9.02774 
9.03327 

9.03881 
9.04434 
9.04986 

533.412 
535.387 
537.368 

639.353 
541.343 
543.338 

545.339 
547.343 
549.353 

2.00912 
2.00995 
2.01078 

2.01160 
2.01242 
2.01325 

2.01407 
2.01489 
2.01571 

4.32853 
4.33031 
4.33208 

4.33386 
4.33563 
4.33741 

4.33918 
4.34095 
4.34271 

9.32553 
9.32936 
9.33319 

9.33702 
9.34084 
9.34466 

9.34847 
9.35229 
9.35610 

.123305 
.123153 
.123001 

.122850 
.122699 
.122549 

.122399 
.122249 
.122100 

8.20 

67.2400 

2.86356 

9.05539 

551.368 

2.01653 

4.34448 

9.35990 

.121951 

8.21 
8.22 
8.23 

8.24 
8.25 
8.26 

8.27 
8.28 
8.29 

67.4041 
67.5684 
67.7329 

67.8976 
68.0625 
68.2276 

68.3929 
68.5584 

68.7241 

2.86531 
2.86705 
2.86880 

2.87054 
2.87228 
2.87402 

2.87576 

2.87750 
2.87924 

9.06091 
9.06642 
9.07193 

9.07744 

9.08295 
9.08845 

9.09.395 
9.09945 
9.10494 

553:388 
555.412 
557.442 

559.476 
561.516 
563.560 

565.609 
567.664 
569.723 

2.01735 
2.01817 
2.01899 

2.01980 
2.02062 
2.02144 

2.02225 
2.02307 
2.02388 

4.34625 
4.34801 
4.34977 

4.35153 
4.35329 
4.35505 

4.35681 
4.35856 
4.36032 

9.36370 
9.36751 
9.37130 

9.37510 
9.37889 
9.38268 

9.38646 
9.39024 
9.39402 

.121803 
.121655 
.121507 

.121359 
.121212 
.121065 

.120919 
.120773 
.120627 

8.30 

68.8900 

2.88097 

9.11043 

571.787 

2.02469 

4.36207 

9.39780 

.120482 

8.31 
8.32 
8.33 

8.34 
8.35 
8.36 

8.37 
8.38 
8.39 

69.0561 
69.2224 
69.3889 

69.5556 
69.7225 
69.8896 

70.0569 
70.2244 
70.3921 

2.88271 
2.88444 
2.88617 

2.88791 
2.88964 
2.89137 

2.89310 
2.89482 
2.8f)655 

9.11592 
9.12140 
9.12688 

9.13236 
9.13783 
9.14330 

9.14877 
9.15423 
9.15969 

573.856 
575.930 
578.010 

580.094 
582.183 
584.277 

586.376 
588.480 
590.590 

2.02551 
2.02632 
2.02713 

2.02794 
2.02875 
2.02956 

2.03037 
2.03118 
2.03199 

4.36382 
4.36557 
4.36732 

4.36907 
4.37081 
4.37256 

4.37430 
4.37604 

4.37778 

9.40157 
9.40534 
9.40911 

9.41287 
9.41663 
9.42039 

9.42414 
9.42789 
9.43164 

.120337 
.120192 

.120048 

.119904 
.119760 
.119617 

.119474 
.119332 
.119190 

8.40 

70.5600 

2.89828 

9.16515 

592.704 

2.03279 

4.37952 

9.43539 

.119048 

8.41 

8.42 
8.43 

8.44 
8.45 
8.46 

8.47 
8.48 
8.49 

70.7281 
70.8964 
71.0649 

71.2336 
71.4025 
71.5716 

71.7409 
71.9104 
72.0801 

2.90000 
2.90172 
2.90345 

2.90517 
2.90689 
2.90861 

2.91033 
2.91204 
2.91376 

9.17061 
9.17606 
9.18150 

9.18695 
9.19239 
9.19783 

9.20326 
9.20869 
9.21412 

594.823 
596.948 
599.077 

601.212 
603.351 
605.496 

607.645 
609.800 
611.960 

2.03360 
2.03440 
2.03521 

2.03601 
2.03682 
2.03762 

2.03842 
2.03923 
2.04003 

4.38126 
4.38299 
4.38473 

4.38646 
4.38819 
4.38992 

4.39165 
4.39338 
4.39510 

9.43913 
9.44287 
9.44661 

9.45034 
9.45407 
9.45780 

9.46152 
9.46525 
9.46897 

.118906 
.118765 
.118624 

.118483 
.118343 
.118203 

.118064 
.117925 
.117786 

8.50 

72.2500 

2.91548 

9.21954 

614.125 

2.04083 

4.39683 

9.47268 

.117647 

n 

n^ 

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VlOn 

^100  n 

%rl] 

Powers  —  Roots  — 

Reciprocals 

109 

n 

n^ 

Vn 

VlOn 

n^ 

</n 

</10n 

1/n 

</100n 

8.50 

72.2500 

2.91548 

9.21954 

614.125 

2.04083 

4.39683 

9.47268 

.117647 

8.51 
8.52 
8.53 

8.54 

8.55 
8.56 

8.57 
8.58 
8.59 

72.4201 
72.5904 
72.7609 

72.9316 
73.1025 
73.2736 

73.4449 
73.6164 

73.7881 

2.91719 
2.91890 
2.92062 

2.92233 
2.92404 
2.92575 

2.92746 
2.92916 
2.93087 

9.22497 
9.23038 
9.23580 

9.24121 
9.24662 
9.25203 

9.25743 
9.26283 
9.26823 

616.295 
618.470 
620.650 

622.836 
625.026 
627.222 

629.423 
631.629 
633.840 

2.04163 
2.04243 
2.04323 

2.04402 
2.04482 
2.04562 

2.04641 
2.04721 
2.04801 

4.39855 
4.40028 
4.40200 

4.40372 
4.40543 
4.40715 

4.40887 
4.41058 
4.41229 

9.47640 
9.48011 
9.48381 

9.48752 
9.49122 
9.49492 

9.49861 
9.50231 
9.50600 

.117509 
.117371 
.117233 

.117096 
.116959 
.116822 

.116686 
.116550 
.116414 

8.60 

73.9600 

2.93258 

9.27362 

636.056 

2.04880 

4.41400 

9.50969 

.116279 

8.61 
8.62 
8.63 

8.64 
8.65 
8.66 

8.67 
8.68 
8.69 

74.1321 
74.3044 
74.4769 

74.6496 
74.8225 
74.9956 

75.1689 
75.3424 
75.5161 

2.93428 
2.93598 
2.93769 

2.93939 
2.94109 
2.94279 

2.94449 
2.94618 

2.94788 

9.27901 
9.28440 

9.28978 

9.29516 
9.30054 
9.30591 

9.31128 
9.31665 
9.32202 

638.277 
640.504 
642.736 

644.973 
647.215 
649.462 

651.714 
653.972 
656.235 

2.04959 
2.05039 
2.05118 

2.05197 
2.05276 
2.05355 

2.05434 
2.05513 
2.05592 

4.41571 
4.41742 
4.41913 

4.42084 
4.42254 
4.42425 

4.42595 
4.42765 
4.42935 

9.51337 
9.51705 
9.52073 

9.52441 

9.52808 
9.53175 

9.53542 
9.53908 
9.54274 

.116144 
.116009 
.115875 

.115741 
.115607 
.115473 

.115340 
.115207 
.115075 

8.70 

75.6900 

2.94958 

9.32738 

658.503 

2.05671 

4.43105 

9.54640 

.114943 

8.71 
8.72 
8.73 

8.74 
8.75 
8.76 

8.77 
8.78 
8.79 

75.8641 
76.0384 
76.2129 

76.3876 
76.5625 
76.7376 

76.9129 
77.0884 
77.2641 

2.95127 
2.95296 
2.95466 

2.95635 
2.95804 
2.95973 

2.96142 
2.96311 
2.96479 

9.33274 
9.33809 
9.34345 

9.34880 
9.35414 
9.35949 

9.36483 
9.37017 
9.37550 

660.776 
663.055 
665.339 

667.628 
669.922 
672.221 

674.526 
676.836 
679.151 

2.05750 
2.05828 
2.05907 

"2.05986 
2.06064 
2.06143 

2.06221 
2.06299 
2.06378 

4.43274 
4.43444 
4.43613 

4.43783 
4.43952 
4.44121 

4.44290 
4.44459 
4.44627 

9.55006 
9.55371 
9.55736 

9.56101 
9.56466 
9.56830 

9.57194 
9.57557 
9.57921 

.114811 
.114679 
.114548 

.114416 
.114286 
.114155 

.114025 
.113895 
.113766 

8.80 

77.4400 

2.96648 

9.38083 

681.472 

2.06456 

4.44796 

9.58284 

.113636 

8.81 
8.82 
8.83 

8.84 
8.85 
8.86 

8.87 
8.88 
8.89 

77.6161 
77.7924 
77.9689 

78.1456 
78.3225 
78.4996 

78.6769 
78.8544 
79.0321 

2.96816 
2.96985 
2.97153 

2.97321 
2.97489 
2.97658 

2.97825 
2.97993 
2.98161 

9.38616 
9.39149 
9.39681 

9.40213 
9.40744 
9.41276 

9.41807 
9.42338 
9.42868 

683.798 
686.129 
688.465 

690.807 
693.154 
695.506 

697.864 
700.227 
702.595 

2.06534 
2.06612 
2.06690 

2.06768 
2.06846 
2.06924 

2.07002 
2.07080 
2.07157 

4.44964 
4.45133 
4.45301 

4.45469 
4.45637 
4.45805 

4.45972 
4.46140 
4.46307 

9.58647 
9.5^)009 
9.59372 

9.59734 
9.60095 
9.60457 

9.60818 
9.61179 
9.61540 

.113507 
.113379 
.113250 

.113122 
.112994 
.112867 

.112740 
.112613 
.112486 

8.90 

79.2100 

2.98329 

9.43398 

704.969 

2.07235 

4.46475 

9.61900 

.112360 

8.91 
8.92 
8.93 

8.94 
8.95 
8.96 

8.97 

.  8.98 

8.99 

79.3881 
79.5664 
79.7449 

79.9236 
80.1025 
80.2816 

80.4609 
80.6404 

80.8201 

2.98496 
2.98664 
2.98831 

2.98998 
2.99166 
2.99333 

2.99500 
2.99666 
2.99833 

9.43928 
9.44458 
9.44987 

9.45516 
9.46044 
9.46573 

9.47101 
9.47629 
9.48156 

707.348 
709.732 
712.122 

714.517 
716.917 
719.323 

721.734 
724.151 
726.573 

2.07313 
2.07390 
2.07468 

2.07545 
2.07622 
2.07700 

2.07777 

2.07854 
2.07931 

4.46642 
4.46809 
4.46976 

4.47142 
4.47309 
4.47476 

4.47642 
4.47808 
4.47974 

9.62260 
9.62620 
9.62980 

9.63339 
9.63698 
9.64057 

9.64415 
9.64774 
9.65132 

.112233 
.112108 
.111982 

.111857 
.111732 
.111607 

.111483 
.111359 
.111235 

900 

81.0000 

3.00000 

9.48683 

729.000 

2.08008 

4.48140 

9.65489 

.111111 

n 

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no 

Powers — Roots  — 

Reciprocals 

[VI 

n 

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1/n 

^100 1* 

9.00 

81.0000 

3.00000 

9.48683 

729.000 

2.08008 

4.48140 

9.65489 

.111111 

.110988 
.110866 
.110742 

.110619 
.110497 
.110375 

.110254 
.110132 
.110011 

9.01 
9.02 
9.03 

9.04 
9.05 
9.06 

9.07 

9.08 
9.09 

81.1801 
81.3604 
81.5409 

81.7216 
81.9025 
82.0836 

82.2649 
82.4464 
82.6281 

3.00167 
3.00333 
3.00500 

3.00666 
3.00832 
3.00998 

3.01164 
3.01330 
3.01496 

9.49210 
9.49737 
9.50263 

9.50789 
9.51315 
9.51840 

9.52365 

9.52890 
9.53415 

731.433 
733.871 
736.314 

738.763 
741.218 
743.677 

746.143 
748.613 
751.089 

2.08086 
2.08162 
2.08239 

2.08316 
2.08393 
2.08470 

2.08546 
2.08623 
2.08699 

4.48306 
4.4847^ 
4.48638 

4.48803 
4.48969 
4.49134 

4.49299 
4.49464 
4.49629 

9.65847 
9.66204 
9.66561 

9.66918 
9.67274 
9.67630 

9.67986 
9.68342 

9.68697 

9.10 

82.8100 

3.01662 

9.53939 

763.571 

2.08776 

4.49794 

9.69052 

.109890 

9.11 
9.12 
9.13 

9.14 
9.15 
9.16 

9.17 
9.18 
9.19 

82.9921 
83.1744 
83.3569 

83.5396 
83.7225 
83.9056 

84.0889 
84.2724 
84.4561 

3.01828 
3.01993 
3.02159 

3.02324 
3.02490 
3.02655 

3.02820 
3.02985 
3.03150 

9.54463 
9.54987 
9.55510 

9.56033 
9.56556 
9.57079 

9.57601 
9.58123 
9.58645 

756.058 
758.551 
761.048 

763.552 
766.061 

768.576 

771.096 
773.621 
776.152 

2.08852 
2.08929 
2.09005 

2.09081 
2.09158 
2.09234 

2.09310 
2.09386 
2.09462 

4.49959 
4.50123 
4.50288 

4.50462 
4.60616 
4.60781 

4.50946 
4.51108 
4.51272 

9.69407 
9.69762 
9.70116 

9.70470 

9.70824 
9.71177 

9.71531 

9.71884 
9.72236 

.109769 
.109649 
.109529 

.109409 
.109290 
.109170 

.109051 
.108932 
.108814 

9.20 

84.6400 

3.03315 

9.59166 

778.688 

2.09538 

4.61436 

9.72589 

.108696 

9.21 
9.22 
9.23 

9.24 
9.25 
9.26 

9.27 
9.28 
9.29 

84.8241 
85.0084 
85.1929 

85.3776 
85.5625 
85.7476 

85.9329 
86.1184 
86.3041 

3.03480 
3.03645 
3.03809 

3.03974 
3.04138 
3.04302 

3.04467 
3.04631 
3.04795 

9.59687 
9.60208 
9.60729 

9.61249 
9.61769 
9.62289 

9.62808 
9.63328 
9.63846 

781.230 
783.777 
786.330 

788.889 
791.453 
794.023 

796.598 
799.179 
801.765 

2.09614 
2.09690 
2.09765 

2.09841 
2.09917 
2.09992 

2.10068 
2a0144 
2.10219 

4.61599 
4.51763 
4.51926 

4.62089 
4.52252 
4.52416 

4.62578 
4.52740 
4.52903 

9.72941 
9.73293 
9.73645 

9.73996 
9.74348 
9.74699 

9.75049 
9.75400 
9.75750 

.108578 
.108460 
.108342 

.108225 
.108108 
.107991 

.107875 
.107759 
.107643 

9.30 

86.4900 

3.04959 

9.64365 

804.357 

2.10294 

4.53065 

9.76100 

.107527 

9.31 
9.32 
9.33 

9.34 
9.35 
9.36 

9.37 
9.38 
9.39 

86.6761 
86.8624 
87.0489 

87.2356 
87.4225 
87.6096 

87.7969 
87.9844 
88.1721 

3.05123 
3.05287 
3.05450 

3.05614 
3.05778 
3.05941 

3.06105 
3.06268 
3.06431 

9.64883 
9.65401 
9.65919 

9.66437 
9.66954 
9.67471 

9.67988 
9.68504 
9.69020 

806.954 
809.558 
812.166 

814.781 
817.400 
820.026 

822.657 
825.294 
827.936 

2.10370 
2.10445 
2.10520 

2.10595 
2.10671 
2.10746 

2.10821 

2.10896 
2.10971 

4.53228 
4.53390 
4.53552 

4.63714 
4.53876 
4.64038 

4.54199 
4.54361 
4.54522 

9.76450 
9.76799 
9.77148 

9.77497 
9.77846 
9.78195 

9.78543 
9.78891 
9.79239 

.107411 
.107296 
.107181 

.107066 
.106952 
.106838 

.106724 
.106610 
.106496 

9.40 

88.3600 

3.06594 

9.69536 

830.584 

2.11046 

4.54684 

9.79586 

.106383 

9.41 
9.42 
9.43 

9.44 
9.45 
9.46 

9.47 
9.48 
9.49 

88.5481 
88.7364 
88.9249 

89.1136 
89.3025 
89.4916 

89.6809 
89.8704 
90.0601 

3.06757 
3.06920 
3.07083 

3.07246 
3.07409 
3.07571 

3.07734 
3.07896 
3.08058 

9.70052 
9.70567 
9.71082 

9.71597 
9.72111 
9.72625 

9.73139 
9.73653 
9.74166 

833.238 
835.897 
838.562 

841.232 
843.909 
846.591 

849.278 
851.971 
854.670 

2.11120 
2.11196 
2.11270 

2.11344 
2.11419 
2.11494 

2.11668 
2.11642 
2.11717 

4.54845 
4.55006 
4.65167 

4.65328 
4.55488 
4.55649 

4.55809 
4.55970 
4.56130 

9.79933 
9.80280 
9.80627 

9.80974 
9.81320 
9.81666 

9.82012 
9.82367 
9.82703 

.106270 
.106157 
.106045 

.105932 
.105820 
.105708 

.105597 
.106485 
.105374 

9.50 

90.2500 

3.08221 

9.74679 

857.376 

2.11791 

4.66290 

9.83048 

.105263 

n 

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Powers  — Roots  — 

Reciprocals 

111 

n 

n^ 

Vn 

VlOn 

n^ 

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^10  n 

1/n 

s/lOOn 

9.50 

90.2500 

3.08221 

9.74679 

857.375 

2.11791 

4.56290 

9.83048 

.105263 

9.51 
9.52 
9.53 

9.54 
9.55 
9.56 

9.57 
9.58 
9.59 

90.4401 
90.6304 
90.8209 

91.0116 
91.2025 
91.3936 

91.5849 
91.7764 
91.9681 

3.08383 
3.08545 
3.08707 

3.08869 
3.09031 
3.09192 

3.09354 
3.09516 
3.09677 

9.75192 
9.75705 
9.76217 

9.76729 
9.77241 
9.77753 

9.78264 
9.78775 
9.79285 

860.085 
862.801 
865.523 

868.251 
870.984 
873.723 

876.467 
879.218 
881.974 

2.11865 
2.11940 
2.12014 

2.12088 
2.12162 
2.12236 

2.12310 
2.12384 
2.12458 

4.56450 
4.56610 
4.56770 

4.56930 
4.57089 
4.57249 

4.57408 
4.57567 

4.57727 

9.83392 
9.83737 
9.84081 

9.84425 
9.84769 
9.85113 

9.85456 
9.85799 
9.86142 

.105152 
.105042 
.104932 

.104822 
.104712 
.104603 

.104493 
.104384 
.104275 

9.60 

92.1600 

3.09839 

9.79796 

884.736 

2.12532 

4.57886 

9.86485 

.104167 

9.61 
9.62 
9.63 

9.64 
9.65 
9.66 

9.67 
9.68 
9.69 

92.3521 
92.5444 
92.7369 

92.9296 
93.1225 
93.3156 

93.5089 
93.7024 
93.8961 

3.10000 
3.10161 
3.10322 

3.10483 
3.10644 
3.10805 

3.10966 
3.11127 
3.11288 

9.80306 
9.80816 
9.81326 

9.81835 
9.82344 
9.82853 

9.83362 
9.83870 
9.84378 

887.504 
8^)0.277 
893.056 

895.841 
898.632 
901.429 

904.231 
907.039 
909.853 

2.12605 
2.12679 
2.12753 

2.12826 
2.12900 
2.12974 

2.13047 
2.13120 
2.13194 

4.58045 
4.58204 
4.58362 

4.68521 
4.58679 
4.58838 

4.58996 
4.59154 
4.59312 

9.86827 
9.87169 
9.87511 

9.87853 
9.88195 
9.88536 

9.88877 
9.89217 
9.89558 

.104058 
.103950 
.103842 

.103734 
.103627 
.103520 

.103413 
.103306 
.103199 

9.70 

94.0900 

3.11448 

9.84886 

912.673 

2.13267 

4.59470 

9.89898 

.103093 

9.71 
9.72 
9.73 

9.74 
9.75 
9.76 

9.77 
9.78 
9.79 

94.2841 
94.4784 
94.6729 

94.8676 
95.0625 
95.2576 

95.4529 
95.6484 
95.8441 

3.11609 
3.11769 
3.11929 

3.12090 
3.12250 
3.12410 

3.12570 
3.12730 
3.12890 

9.85393 
9.85901 
9.86408 

9.86914 
9.87421 
9.87927 

9.88433 
9.88939 
9.89444 

915.499 
918.330 
921.167 

924.010 
926.859 
929.714 

932.575 
935.441 
938.314 

2.13340 
2.13414 
2.13487 

2.ia560 
2.13633 
2.13706 

2.13779 
2.13852 
2.13925 

4.59628 
4.59786 
4.59943 

4.60101 
4.60258 
4.60416 

4.60573 
4.60730 

4.60887 

9.90238 
9.90578 
9.90918 

9.91257 
9.91596 
9.91935 

9.92274 
9.92612 
9.92950 

.102987 
.102881 
.102775 

.102669 
.102564 
.102459 

.102.354 
.102249 
.102145 

9.80 

96.0400 

3.13050 

9.89949 

941.192 

2.13997 

4.61044 

9.93288 

.102041 

9.81 

9.82 
9.83 

9.84 
9.85 
9.86 

9.87 
9.88 
9.89 

96.2361 
96.4324 
96.6289 

96.8256 
97.0225 
97.2196 

97.4169 
97.6144 
97.8121 

3.13209 
3.13369 
3.13528 

3.13688 
3.13847 
3.14006 

3.14166 
3.14325 
3.14484 

9.90454 
9.90959 
9.91464 

9.91968 
9.92472 
9.92975 

9.93479 
9.93982 
9.94485 

944.076 
946.966 
949.862 

952.764 
955.672 
958.585 

961.505 
964.430 
967.362 

2.14070 
2.14143 
2.14216 

2.14288 
2.14361 
2.14433 

2.14506 
2.14578 
2.14651 

4.61200 
4.61357 
4.61514 

4.61670 
4.61826 
4.61983 

4.62139 
4.62295 
4.62451 

9.93626 
9.93964 
9.94301 

9.94638 
9.94975 
9.95311 

9.95648 
9.95984 
9.96320 

.101937 
.101833 
.101729 

.101626 
.101523 
.101420 

■  .101317 
.101215 
.101112 

9.90 

9.91 
9.92 
9.93 

9.94 
9.95 
9.96 

9.97 
9.98 
9.99 

98.0100 

3.14643 

9.94987 

970.299 

2.14723 

4.62607 

9.96655 

.101010 

98.2081 
98.4064 
98.6049 

98.8036 
99.0025 
99.2016 

99.4009 
99.6004 
99.8001 

3.14802 
3.14960 
3.15119 

3.15278 
3.15436 
3.15595 

3.15753 
3.15911 
3.16070 

9.95490 
9.95992 
9.96494 

9.96995 
9.97497 
9.97998 

9.98499 
9.98999 
9.99500 

973.242 
976.191 
979.147 

982.108 

985.075 
988.048 

991.027 
994.012 
997.003 

2.14795 
2.14867 
2.14940 

2.15012 
2.15084 
2.15156 

2.15228 
2.15300 
2.15372 

4.62762 
4.62918 
4.63073 

4.6.3229 
4.63384 
4.63539 

4.63694 
4.63849 
4.64004 

9.96991 
9.97326 
9.97661 

9.97996 
9.98331 
9.98665 

9.98999 
9.99333 
9.99667 

.100908 
.100806 
.100705 

.100604 
.100503 
.100402 

.100301 
.100200 
.100100 

10.00 

100.000 

3.16228 

10.0000 

1000.00 

2.15443 

4.64159 

10.0000 

.100000 

n 

n2 

Vn 

VlOri 

n^ 

^n 

^10  n 

1/n 

^100  w 

112       Table  Til  —  Napierian  or  Natural  Logarithms        [vii 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

0.0 

5.395 

6.088 

6.493 

6.781 

7.004 

7.187 

7.341 

7.474 

7.592 

0.1 
0.2 
0.3 

=  7.697 
1  8.391 
§  8.796 

7.793 
8.439 
8.829 

7.880 
8.486 
8.861 

7.960 
8.530 
8.891 

8.034 
8.573 
8.921 

8.103 
J?.8l4 
8.950 

8.167 
8.653 
8.978 

8.228 
8.691 
9.006 

8.285 
8.727 
9.032 

8.339 
8.762 
9.058 

0.4 
0.5 
0.6 

>   9.084 
U   9.307 
1  9.489 

9.108 
9.327 
9.506 

9.132 
9.346 
9.522 

9.156 
9.365 
9.538 

9.179 
9.384 
9.554 

9.201 
9.402 
9.569 

9.223 
9.420 
9.584 

9.245 
9.438 
9.600 

9.266 
9.455 
9.614 

9.287 
9.472 
9.629 

0.7 
0.8 
0.9 

f  9.643 
-^  9.777 
H  9.895 

9.658 
9.789 
9.906 

9.671 
9.802 
9.917 

9.685 
9.814 
9.927 

9.699 
9.826 
9.938 

9.712 
9.837 
9.949 

9.726 
9.849 
9.959 

9.739 
9.861 
9.970 

9.752 

9.872 
9.980 

9.764 
9.883 
9.990 

1.0 

0.00000 

0995 

1980 

2956 

3922 

4879 

5827 

6766 

7696 

8618 

1.1 
1.2 
1.3 

9531 
0.1  8232 
0.2  6236 

*0436 
9062 
7003 

*1333 
9885 
7763 

*2222 

*0701 

8518 

*3103 

*1511 

9267 

*3976 
*2314 
*0010 

*4842 
*3111 
*0748 

*5700 
*3t)02 
*1481 

*6551 
*4686 
*2208 

*7395 
*5464 
*2930 

1.4 
1.5 
1.6 

0.3  3647 

0.4  0547 

7000 

4359 
1211 
7623 

5066 
1871 
8243 

5767 
2527 
8858 

6464 
3178 
9470 

7156 

3825 

*0078 

7844 

4469 

*0682 

8526 

5108 

*1282 

9204 

5742 
*1879 

9878 

6373 

*2473 

1.7 
1.8 
1.9 

0.5  3063 

8779 

0.6  4185 

3649 
9333 
4710 

4232 

9884 
5233 

4812 

*0432 

5752 

5389 

*0977 

6269 

5962 

*1519 

6783 

6531 

*2058 

7294 

7098 

*2594 

7803 

7661 

*3127 

8310 

8222 

*3658 

8813 

2.0 

9315 

9813 

*0310 

*0804 

*1295 

*1784 

*2271 

*2755 

*3237 

*371G 

2.1 
2.2 
2.3 

0.7  4194 

8846 

0.8  3291 

4669 
9299 
3725 

5142 
9751 
4157 

5612 

*0200 

4587 

6081 

*0648 

5015 

6547 

*1093 

5442 

7011 

*1536 

5866 

7473 

*1978 
6289 

7932 

*2418 

6710 

8390 

*2855 

7129 

2.4 
2.5 
2.6 

7547 

0.9  1629 

5551 

7963 
2028 
5935 

8377 
2426 
6317 

8789 
2822 
6698 

9200 
3216 
7078 

9609 
3609 
7456 

*0016 
4001 
7833 

*0422 
4391 

8208 

*0826 
4779 

8582 

*1228 
5166 

8954 

2.7 
2.8 
2.9 

9325 

1.0  2962 

6471 

9695 
3318 
6815 

*0063 
3674 
7158 

*0430 
4028 
7500 

*0796 
4380 

7841 

*1160 
4732 
8181 

*1523 

5082 
8519 

*1885 
5431 
8856 

*2245 
5779 
9192 

*2604 
6126 
9527 

3.0 

9861 

*0194 

*0526 

*0856 

*1186 

*1514 

*1841 

*2168 

*2493 

*2817 

3.1 
3.2 
3.3 

1.1  3140 
6315 
9392 

3462 
6627 
9695 

3783 
6938 
9996 

4103 

7248 

*0297 

4422 

7557 

*0597 

4740 

7865 
*0896 

5057 

8173 

*1194 

5373 

8479 

*1491 

5688 

8784 

*1788 

6002 

9089 

*2083 

3.4 
3.5 
3.6 

1.2  2378 
5276 
8093 

2671 
5562 
8371 

2964 
5846 
8647 

3256 
6130 
8923 

3547 
6413 
9198 

3837 
6695 
9473 

4127 
6976 
9746 

4415 

7257 

*0019 

4703 

7536 

*0291 

4990 

7815 

*0563 

3.7 
3.8 
3.9 

1.3  0833 
3500 
6098 

1103 
3763 
6354 

1372 
4025 
6609 

1641 

4286 
6864 

1909 
4547 
7118 

2176 

4807 
7372 

2442 
5067 
7624 

2708 
5325 

7877 

2972 
5584 
8128 

3237 
5841 
8379 

4.0 

8629 

8879 

9128 

9377 

9624 

9872 

*0118 

*0364 

*0610 

*0854 

4.1 
4.2 
4.3 

1.4 1099 
3508 
5862 

1342 
3746 
6094 

1585 
3984 
6326 

1828 
4220 
6557 

2070 
4456 

6787 

2311 
4692 
7018 

2552 
4927 

7247 

2792 
5161 
7476 

3031 
5395 
7705 

3270 
5629 
7933 

4.4 
4.5 
4.6 

8160 

1.5  0408 

2606 

8387 
0630 
2823 

8614 
0851 
3039 

8840 
1072 
3256 

9065 
1293 
3471 

9290 
1513 
3687 

9515 
1732 
3902 

9739 
1951 
4116 

9962 
2170 
4330 

*0185 
2388 
4543 

4.7 

4.8 
4.9 

4756 
6862 
8924 

4969 
7070 
9127 

5181 

7277 
9331 

5393 

7485 
9534 

5604 
7691 
9737 

5814 

7898 
9939 

6025 

8104 

*0141 

6235 

8309 

*0342 

6444 

8515 

*0543 

6653 

8719 

*0744 

5.0 

1.6  0944 

1144 

1343 

1542 

1741 

1939 

2137 

2334 

2531 

2728 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

vir\ 

Napierian 

or  Natural 

Logarithms 

113 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

5.0 

1.6  0944 

1144 

1343 

1542 

1741 

1939 

2137 

2334 

2531 

2728 

5.1 
5.2 
5.3 

2924 
4866 
6771 

3120 

5058 
6959 

3315 
5250 
7147 

3511 
5441 
7335 

3705 
6632 
7523 

3900 
5823 
7710 

4094 
6013 

7896 

4287 
6203 

8083 

4481 
6393 
8269 

4673 
6582 
8455 

5.4 
5.5 
5.6 

8640 
1.7  0475 

2277 

8825 
0;)56 
2455 

9010 
0838 
2633 

9194 
1019 
2811 

9378 
1199 

2988 

9562 
1380 
3166 

9745 
1560 
3342 

9928 
1740 
3519 

*0111 
1919 
3695 

*0293 
2098 
3871 

5.7 
5.8 
5.9 

4047 
5786 
7495 

4222 

5958 
7665 

4397 
6130 

7834 

4572 
6302 
8002 

4746 
6473 
8171 

4920 
6644 
8339 

6094 
6815 

8507 

6267 
6985 
8675 

6440 
7156 

8842 

6613 
7326 
9009 

6.0 

9176 

9342 

9509 

9(575 

.  9840 

*0006 

*0171 

*0336 

*0500 

*0665 

6.1 
6.2 
6.3 

1.80829 
2455 
4055 

0993 
2616 
4214 

1156 
2777 
4372 

1319 
2938 
4530 

1482 
3098 

4688 

1645 
3258 
4845 

1808 
3418 
5003 

1970 

3578 
5160 

2132 

3737 
5317 

2294 
3896 
5473 

6.4 
6.5 
6.6 

5630 

7180 
8707 

5786 
7334 
8858 

5942 
7487 
9010 

6097 
7641 
9160 

6253 

7794 
9311 

6408 
7947 
9462 

6563 
8099 
9612 

6718 
8251 
9762 

6872 
8403 
9912 

7026 

8556 

*0061 

6.7 
6.8 
6.9 

7.0 

1.90211 
1692 
3152 

0360 
1839 
3297 

0509 
1986 
3442 

0658 
2132 

3586 

0806 
2279 
3730 

0954 
2425 
3874 

1102 
2571 
4018 

1250 
2716 
4162 

1398 
2862 
4305 

1645 

3007 
4448 

4501 

4734 

4876 

5019 

5161 

5303 

5445 

5586 

5727 

6869 

7.1 

7.2 
7.3 

6009 
7408 
8787 

6150 
7547 
8924 

6291 
7685 
9061 

6431 
7824 
9198 

6571 
7962 
9334 

6711 
8100 
9470 

6851 
8238 
9606 

6991 
8376 
9742 

7130 
8513 

9877 

7269 

8650 

*0013 

7.4 
7.5 
7.6 

2.00148 
1490 

2815 

0283 
1624 
2946 

0418 
1757 
3078 

0553 
1890 
3209 

0687 
2022 
3340 

0821 
2155 
3471 

0956 

2287 
3601 

1089 
2419 
3732 

1223 
2551 
3862 

1367 
2683 
3992 

7.7 
7.8 
7.9 

4122 
5412 
6686 

4252 
5540 
6813 

4381 
5668 
6939 

4511 
5796 
7065 

4640 
5924 
7191 

4769 
6051 
7317 

4898 
6179 
7443 

5027 
6306 

7568 

5156 
6433 
7694 

6284 
6560 
7819 

8.0 

7944 

8069 

8194 

8318 

8443 

8567 

8691 

8815 

8939 

9063 

8.1 
8.2 
8.3 

9186 

2.10413 

1626 

9310 
0535 
1746 

9433 
0657 
1866 

9556 
0779 
1986 

9679 
0900 
2106 

9802 
1021 
2226 

9924 
1142 
2346 

*0047 
1263 
2465 

*0169 
1384 
2585 

*0291 
1505 
2704 

8.4 
8.5 
8.6 

2823 
4007 
5176 

2942 
4124 
5292 

3061 
4242 
5409 

3180 
4359 
5524 

3298 
4476 
5640 

3417 
4593 
5756 

3535 
4710 

5871 

3653 

4827 
5987 

3771 
4943 
6102 

3889 
5060 
6217 

8.7 
8.8 
8.9 

6332 
7475 
8605 

6447 

7589 
8717 

6562 

7702 
8830 

6677 
7816 
8942 

6791 
7929 
9054 

6905 
8042 
9165 

7020 
8155 
9277 

7134 
8267 
9389 

7248 
8380 
9500 

7361 
8493 
9611 

9.0 

9722 

9834 

9944 

*0055 

*0166 

*0276 

*0387 

*0497 

*0607 

*0717 

9.1 
9.2 
9.3 

2.2  0827 
1920 
3001 

0937 
2029 
3109 

1047 
2138 
3216 

1157 
2246 
3324 

1266 
2354 
3431 

1375 
2462 
3538 

1485 
2570 
3645 

1594 
2678 
3751 

1703 

2786 
3858 

1812 
2894 
3965 

9.4 
9.5 
9.6 

4071 
6129 
6176 

4177 
5234 
6280 

4284 
5339 
6384 

4390 
5444 
6488 

4496 
5549 
6592 

4601 
5654 
6696 

4707 
5759 
6799 

4813 
5863 
6903 

4918 
5968 
7006 

6024 
6072 
7109 

9.7 
9.8 
9.9 

7213 

8238 
9253 

7316 
8340 
9354 

7419 
8442 
9455 

7521 
8544 
9556 

7624 
8646 
9657 

7727 
8747 
9757 

7829 
8849 
9858 

7932 
8950 
9958 

8034 

9051 

*0058 

8136 

9152 

*0158 

10.0 

2.30259 

0358 

0458 

0558 

0658 

0757 

0857 

0956 

1055 

1154 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

114 

Napierian  or  Natural  Logarithms  — 

-10  to  99 

[VII 

10 

2.30259 

25 

3.21888 

40 

3.68888 

55 

4.00733 

70 

4.24850 

85 

4.44265 

11 

12 
13 
14 

2.39790 
2.48491 
2.56495 
2.63906 

26 

27 
28 
29 

3.25810 
3.29584 
3.33220 
3.36730 

41 
42 
43 
44 

3.71357 
3.73767 
3.76120 
3.78419 

56 

57 
68 
59 

4.02535 
4.04305 
4.06044 
4.07764 

71 

72 
73 
74 

4.26268 
4.27667 
4.29046 
4.30107 

86 

87 
88 
89 

4.45435 
4.46591 
4.47734 

4.48864 

4.49981 

15 

2.70805 

30 

3.40120 

45 

3.80666 

60 

4.09434 

75 

4.31749 

90 

16 
17 
18 
19 

2.77259 
2.83321 
2.89037 
2.94444 

31 
32 
33 
34 

3.43399 
3.46574 
3.49651 
3.5263e 

46 
47 
48 
49 

3.82864 
3.85015 
3.87120 
3.89182 

61 
62 
63 
64 

4.11087 
4.12713 
4.14313 

4.15888 

76 
77 
78 
79 

4.33073 
4.34381 
4.35671 
4.36945 

91 
92 
93 
94 

4.51086 
4.52179 
4.53260 
4.54329 

20 

2.99573 

35 

3.55535 

50 

3.91202 

65 

4.17439 

80 

4.38203 

95 

4.55388 

21 
22 
23 
24 

3.04452 
3.09104 
3.1:3549 
3.17805 

36 
37 
38 
39 

3.58352 
3.61092 
3.63759 
3.66356 

51 
52 
63 
54 

3.93183 
3.95124 
3.97029 
3.98898 

6ij 
67 
68 
69 

4.18965 
4.20469 
4.21951 
4.23411 

81 
82 
83 
84 

4.39445 
4.40672 
4.41884 
4.43082 

96 
97 

98 
99 

4.56435 
4.57471 
4.68497 
4.69512 

NAPIERIAN  ( 

OR   NATURAL  LOGARITHMS  — 

100  TO  409 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

4.6  0517 

1612 

2497 

3473 

4439 

6396 

6344 

7283 

8213 

9135 

11 
12 
13 

4.7  0048 
8749 

4.8  6753 

0953 
9579 
7520 

1850 

*0402 

8280 

2739 

*1218 

9036 

3620 

*2028 

9784 

4493 
*2831 
*0527 

5359 
*3628 
*1265 

6217 
*4419 
*1998 

7068 
*5203 
*2725 

7912 
*6981 
*3447 

14 
15 
16 

4.9  4164 

5.0  1064 

7517 

4876 
1728 
8140 

6683 
2388 
8760 

6284 
3044 
9376 

6981 
3695 
9987 

7673 

4343 

*0695 

8361 

4986 

*1199 

9043 

5625 

*1799 

9721 

6260 

*2396 

*0395 

6890 

*2990 

17 
18 
19 

5.1 3580 

9296 

5.2  4702 

4166 
9860 
6227 

4749 

*0401 

6760 

6329 

*0949 

6269 

6906 

*1494 

6786 

6479 

*2036 

7300 

7048 

*2575 
7811 

7616 

*3111 

8320 

8178 

*3644 

8827 

8739 

*4176 

9330 

20 

9832 

*0330 

*0827 

*1321 

*1812 

*2301 

*2788 

*3272 

*3754 

*4233 

21 
22 
23 

6.34711 

9363 

5.4  3808 

6186 
9816 
4242 

6659 

*0268 

4674 

6129 

*0717 

6104 

6698 

*1165 

6532 

7064 

*1610 

6959 

7628 

*2053 

6383 

7990 

*2495 

6806 

8450 
*2935 

7227 

8907 

*3372 

7646 

24 
25 
26 

8064 

5.5  2146 

6068 

8480 
2545 
6452 

8894 
2943 
6834 

9306 
3339 
7215 

9717 
3733 
7595 

*0126 
4126 
7973 

*0533 
4518 
8350 

*0939 
4908 
8726 

-1343 
6296 
9099 

*1745 
5683 
9471 

27 
28 
29 

9842 
5.6  3479 

6988 

*0212 
3835 
7332 

*0680 
4191 
7675 

*0947 
4545 
8017 

*1313 

4897 
8358 

*1677 
6249 
8698 

*2040 
5599 
9036 

*2402 
5948 
9373 

*2762 
6296 
9709 

*3121 

6643 

*0044 

30 

5.7  0378 

0711 

1043 

1373 

1703 

2031 

2359 

2685 

3010 

3334 

31 
32 
33 

3667 
6832 
9909 

3979 
7144 

*0212 

4300 

7455 

*0513 

4620 

7765 

*0814 

4939 

8074 

*1114 

5267 

8383 

*1413 

6574 

8690 

*1711 

68<)0 

89f)6 

*2008 

6206 

9301 

*2305 

6519 

9606 

*2600 

34 
35 
36 

5.8  2895 
6793 
8610 

3188 
6079 

8888 

3481 
6363 
9164 

3773 
6647 
9440 

4064 
6930 
9715 

4354 
7212 
9990 

4644 

7493 

*0263 

4932 

7774 
*0636 

6220 

8053 

*0808 

6607 

8332 

*1080 

37 
38 
39 

5.9 1360 
4017 
6616 

1620 
4280 
6871 

1889 
4542 
7126 

2158 
4803 
7381 

2426 
6064 
7636 

2693 
5324 

7889 

2969 
6584 
8141 

3226 
6842 
8394 

3489 
6101 
8645 

3764 
6368 
8896 

40 

9146 

9396 

9646 

9894 

*0141 

*0389 

*0635 

*0881 

*1127 

*1372 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Above  409,  use  the  formula     log*  10  ?i  =  loge  n  +  loge  10  =  loge  n  +  2.30268509, 
or  the  formula  log*  n  =  loge  10  •  logio  ^  =  2.30258509  logio  n. 


Table  VIII  —  Multiples  of  M  and  of  1/M 


115 


N 

N'M 

N 

N'M 

0 

1 
2 
3 

4 
5 
6 

7 
8 
9 

10 

11 
12 
13 

14 
15 
16 

17 

18 
19 

20 

21 
22 
23 

24 
25 
26 

27 
28 
29 

30 

31 
32 
33 

34 
35 
36 

37 
38 
39 

0.00000  000 

50 

21.71472  410 

0.43429  448 
0.86858  896 
1.30288  345 

1.73717  793 
2.17147  241 
2.60576  689 

3.04006  137 
3.47435  586 
3.90865  034 

51 
52 
53 

54 
55 
50 

57 

58 
59 

22.14901  858 
22.58331  306 
23.01760  754 

23.45190  202 
23.88619  650 
24.32049  099 

24.75478  547 
25.18907  995 
25.62337  443 

4.34294  482 

60 

26.05766  891 

4.77723  930 
5.21153  378 
5.64582  826 

6.08012  275 
6.51441  723 
6.94871  171 

7.38300  619 
7.81730  067 
8.25159  516 

61 
62 
63 

64 
65 
66 

67 

68 
69 

26.49196  340 
26.92625  788 
27.36055  236 

27.79484  684 
28.22914  132 
28.66343  581 

29.09773  029 
29.53202  477 
29.96631  925 

8.68588  964 

70 

30.40061  373 

9.12018  412 
9.55447  860 

9.98877  308 

10.42306  757 
10.85736  205 
11.29165  653 

11.72595  101 
12.16024  549 
12.59453  998 

71 

72 
73 

74 
75 
76 

77 
78 
79 

30.83490  822 
31.26920  270 
31.70349  718 

32.13779  166 
32.57208  614 
33.00638  062 

38.44067  511 
33.87496  959 
34.30926  407 

13.02883  446 

80 

34.74355  855 

13.46312  894 
13.89742  342 
14.33171  790 

14.76601  238 
15.20030  687 
15.63460  135 

16.06889  583 
16.50319  031 
16.93748  479 

81 
82 
83 

84 
85 
86 

87 
88 
89 

35.17785  303 
35.61214  752 
36.04644  200 

36.48073  648 
36.91503  096 
37.31932  644 

37.78361  993 
38.21791  441 
38.65220  889 

40 

41 
42 
43 

44 
45 
46 

47 
48 
49 

17.37177  928 

90 

39.08650  337 

17.80607  376 
18.24036  824 
18.67466  272 

19.10895  720 
19.54325  169 
19.97754  617 

20.41184  065 
20.84613  513 
21.28042  961 

91 
92 
93 

94 
95 
96 

97 

98 
99 

39.52079  785 
39.95509  234 
40.38938  682 

40.82368  130 
41.25797  578 
41.69227  026 

42.12656  474 
42.56085  923 
42.99515  371 

60 

21.71472  410 

100 

43.42944  819 

A^ 

N-i-M 

N 

N-^M 

0 

1 
2 

3 

4 
5 
6 

7 
8 
9 

10 

11 
12 
13 

14 
15 
16 

17 

18 
19 

20 

21 
22 
23 

24 
25 
26 

27 
28 
29 

30 

31 
32 
33 

34 
35 
36 

37 

38 
39 

0.00000  000 

50 

115.12925  465 

2.30258  509 
4.60517  019 
6.90775  528 

9.21034  037 
11.51292  546 
13.81551  056 

16.11809  565 
18.42068  074 
20.72326  584 

51 
52 
63 

54 
65 
66 

57 
58 
59 

117.43183  974 
119.73442  481 
122.03700  993 

124.33959  502 
126.64218  Oil 
128.94476  621 

131.24735  030 
133.54993  639 
135.85252  049 

23.02585  093 

60 

138.15510  558 

25.32843  602 
27.63102  112 
29.93360  621 

32.23619  130 
34.53877  639 
36.84136  149 

39.14394  658 
41.44653  167 
43.74911  677 

61 
62 
63 

64 
65 
66 

67 
68 
69 

140.45769  067 
142.76027  577 
145.06286  086 

147.36544  595 
149.66803  104 
151.97061  614 

154.27320  123 
156,57578  632 
158.87837  142 

46.05170  186 

70 

161.18095  651 

48.35428  695 
50.65687  205 
62.95945  714 

65.26204  223 
67.56462  732 
59.86721  242 

62.16979  751 
64.47238  260 
66.77496  770 

71 
72 
73 

74 
75 
76 

77 
78 
79 

163.48354  160 
165.78612  670 
168.08871 179 

170.39129  688 
172.69388  197 
174.99646  707 

177.29905  216 
179.60163  725 
181.90422  235 

69.07755  279 

80 

184.20680  744 

71.38013  788 
73.68272  298 
75.98530  807 

78.28789  316 
80.59047  825 
82.89306  335 

85.19564  844 
87.49823  353 
89.80081  863 

81 
82 
83 

84 

85 
86 

87 
88 
89 

186.50939  253 
188.81197  763 
191.11456  272 

193.41714  781 
195.71973  290 
198.02231  800 

200.32490  309 
202.62748  818 
204.93007  328 

40 

41 
42 
43 

44 
45 
46 

47 
48 
49 

92.10340  372 

90 

207.23265  837 

91.40598  881 
96.70857  391 
99.01115  900 

101.31374  409 
103.61632  918 
105.91891  428 

108.22149  937 
110.52408  446 
112.82666  956 

91 
92 
93 

94 
95 

96 

97 
98 
99 

209.53524  346 
211.83782  856 
214.14041  365 

216.44299  874 
218.74558  383 
221.04816  893 

223.35075  402 
225.65333  911 
227.95592  421 

50 

115.12925  465 

100 

230.25850  930 

M=  log^^e  =  .43429  44819  03261  82765 


-^  =  log  10  =  2.30258  50929  94045  68402 
if    ^«  ^ 

log^n  =  log^^n  .  log^lO  =  j^  los,,n. 


116      Table  IX — Logarithms  of  Hyperbolic  Functions 


a? 

Value   Logio 

6- 
Value 

Sinhcc 

Value   Logio 

Coshac 

Value   Logio 

Tanhi» 

Value 

0.00 

1.0000 

.00000 

1.0000 

0.0000 

—  00 

1.0000 

.00000 

.00000 

0.01 
0.02 
0.03 

1.0101 
1.0202 
1.0305 

.00434 
.00869 
.01303 

.99005 
.98020 
.97045 

0.0100 
0.0200 
0.0300 

.00001 
.30106 
.47719 

1.0001 
1.0002 
1.0005 

.00002 
.00009 
.00020 

.01000 
.02000 
.02999 

0.04 
0.05 
0.06 

1.0408 
1.0513 
1.0618 

.01737 
.02171 
.02606 

.96079 
.95123 
.94176 

0.0400 
0.0500 
00600 

.60218 
.69915 

.77841 

1.0008 
1.0013 
1.0018 

.00035 
.00054 
.00078 

.03998 
.04996 
.05993 

0.07 
0.08 
0.09 

1.0725 
1.0833 
1.0942 

.03040 
.03474 
.03909 

.93239 
.92312 
.91393 

0.0701 
0.0801 
0.0901 

.84545 
.90355 
.95483 

1.0025 
1.0032 
1.0041 

.00106 
.00139 
.00176 

.06989 
.07983 
.08976 

0.10 

1.1052 

.04343 

.90484 

0.1002 

.00072 

1.0050 

.00217 

.09967 

0.11 
0.12 
0.13 

1.1163 
1.1275 
1.1388 

.04777 
.05212 
.05646 

.89583 
.88692 
.87809 

0.1102 
0.1203 
0.1304 

.04227 
.08022 
.11517 

1.0061 
1.0072 
1.0085 

.00262 
.00312 
.00366 

.10956 
.11943 
.12927 

0.14 
0.15 
0.16 

1.1503 
1.1618 
1.1735 

.06080 
.06514 
.06949 

.86936 
.86071 
.85214 

0.1405 
0.1506 
0.1607 

.14755 

.17772 
.20597 

1.0098 
1.0113 
1.0128 

.00424 
.00487 
.00554 

.13909 
.14889 
.15865 

0.17 
0.18 
0.19 

1.1853 
1.1972 
1.2092 

.07383 
.07817 
.08252 

.84366 
.83527 
.82696 

0.1708 
0.1810 
0.1911 

.23254 
.25762 
.28136 

1.0145 
1.0162 
1.0181 

.00625 
.00700 
.00779 

.16838 
.17808 

.18775 

0.20 

1.2214 

.08686 

.81873 

0.2013 

.30392 

1.0201 

,00863 

.19738 

0.21 
0.22 
0.23 

1.2337 
1.2461 
1.2586 

.09120 
.09554 
.09989 

.81058 
.80252 
.79453 

0.2115 
0.2218 
0.2320 

.32541 
.34592 
.36555 

1.0221 
1.0243 
1.0266 

.00951 
.01043 
.01139 

.20697 
.21652 
.22603 

0.24 
0.25 
0.26 

1.2712 

1.2840 
1.2969 

.10423 

.10857 
.11292 

.78663 

.77880 
.77105 

0.2423 
0.2526 
0.2629 

.38437 
.40245 
.41986 

1.0289 
1.0314 
1.0340 

.01239 
.01343 
.01452 

.23550 
.24492 
.25430 

0.27 
0.28 
0.29 

1.3100 
1.3231 
1.3364 

.11726 
.12160 
.12595 

.76338 

.75578 

.74826 

0.2733 

0.2837 
0.2941 

.43663 
.45282 
.46847 

1.0367 
1.0395 
1.0423 

.01564 
.01681 
.01801 

.26362 
.27291 
.28213 

0.30 

1.3499 

.13029 

.74082 

0.3045 

.48362 

1.0453 

.01926 

.29131 

0.31 
0.32 
0.33 

1.3634 
1.3771 
1.3910 

.13463 
.13897 
.14332 

.73345 
.72015 
.71892 

0.3150 
0.3255 
0.3360 

.49830 
.51254 
.52637 

1.0484 
1.0516 
1.0549 

.02054 
.02107 
.02323 

.30044 
.30951 
.31852 

0.34 
0.35 
0.36 

1.4049 
1.4191 
1.4333 

.14766 
.15200 
.15635 

.71177 
.70469 
.69768 

0.3466 
0.3572 
0.3678 

.53981 
.55290 
.56564 

1.0584 
1.0619 
1.0655 

.02463 
.02607 
.02755 

.32748 
.33638 
.34521 

0.37 
0.38 
0-39 

1.4477 
1.4623 
1.4770 

.16069 
.16503 
.16937 

.69073 

.68386 
.67706 

.67032 

0.3785 
0.3892 
0.4000 

.57807 
.59019 
.60202 

1.0692 
1.0731 
1.0770 

.02907 
.03063 
.03222 

.35399 
.36271 
.37136 

0.40 

1.4918 

.17372 

0.4108 

.61358 

1.0811 

.03385 

.37995 

0.41 

0.42 
0.43 

1.5068 
1.5220 
1.5373 

.17806 
.18240 
.18675 

.66365 
.65705 
.65051 

0.4216 
0.4325 
0.4434 

.62488 
.63594 
.64677 

1.0852 
1.0895 
1.0939 

.03552 
.03723 
.03897 

.38847 
.39693 
.40532 

0.44 
0.45 
0.46 

1.5527 
1.5683 
1.5841 

.19109 
.19543 
.19978 

.64404 
.63763 
.63128 

0.4543 
0.4653 
0.4764 

.65738 
.66777 
.67797 

1.0984 
1.1030 
1.1077 

.04075 
.04256 
.04441 

.41364 
.42190 
.43008 

0.47 
0.48 
0.49 

1.6000 
1.6161 
1.6323 

.20412 
.20846 
.21280 

.62500 
.61878 
.61263 

0.4875 
0.4986 
0.5098 

.68797 
.69779 
.70744 

1.1125 
1.1174 

1:1225 

.04630 
.04822 
.05018 

.43820 
.44624 
.45422 

050 

1.6487 

.21715 

.60653 

0.5211 

.71692 

1.1276 

.05217 

.46212 

Values  and  Logarithms  of  Hyperbolic  Functions       117 


X 

Value 

Log.o 

Value 

Sin}] 

Value 

I  a? 
Logio 

Cosl 

Value 

Logio 

Tanha? 

Value 

0.50 

1.6487 

.21715 

.60653 

0.5211 

.71692 

1.1276 

.05217 

.46212 

.46995 
.47770 
.48538 

0.51 
0.52 
0.53 

1.6653 
16820 
1.6989 

.22149 
.22583 
.23018 

.60050 
.59452 

.58860 

0.5324 
0.5438 
0.5552 

.72624 
.73540 
.74442 

1.1329 
1.1383 
1.1438 

.05419 
.05625 
.05834 

0.54 
0.55 
0.56 

1.7160 
1.7333 
1.7507 

.23452 
.23886 
.24320 

.58275 
.57695 
.57121 

0.5666 
0.5782 
0.5897 

,75330 
.76204 
.77065 

1.1494 
1.1551 
1.1609 

.06046 
.06262 
.06481 

.49299 
.50052 
.50798 

0.57 
0.58 
C.59 

1.7683 
1-7860 
1.8040 

.24755 
.25189 
.25623 

.56553 
.55990 
.55433 

0.6014 
0.6131 
0.6248 

.77914 

.78751 
.79576 

1.1669 
1.1730 
1.1792 

.06703 
.06929 
.07157 

.51536 
.52267 
.52990 

0.60 

1.8221 

.26058 

.54881 

0.6367 

.80390 

1.1855 

.07389 

.53705 

0.61 
0.62 
0.63 

1.8404 

1.8589 
1.8776 

.26492 
.26926 
.27361 

.54335 
.53794 
.53259 

0.6485 
0.6605 
0.6725 

.81194 
.81987 
.82770 

1.1919 
1.1984 
1.2051 

.07624 
.07861 
.08102 

.54413 
.55113 
.55805 

0.64 
0.65 
0.66 

1.8965 
1.9155 
1.9348 

.27795 
.28229 
.28664 

.52729 
.52205 
.51685 

0.6846 
0.6967 
0.7090 

.83543 
.84308 
.85063 

1.2119 
1.2188 
1.2258 

.08346 
.08593 
.08843 

.56490 
.57167 
.57836 

0.67 
0.68 
0.69 

1.9542 
1.9739 
1.9937 

.29098 
.29532 
.29966 

.51171 
.50662 
.50158 

0.7213 
0.7336 
0.7461 

.85809 
.86548 
.87278 

1.2330 
1.2402 
1.2476 

.09095 
.09351 
.09609 

.58498 
.59152 
.59798 

0.70 

2.0138 

.30401 

.49659 

0.7586 

.88000 

1.2552 

.09870 

.60437 

0.71 
0.72 
0.73 

2.0340 
2.0544 
2.0751 

.30835 
.31269 
.31703 

.49164 
.48675 
.48191 

0.7712 
0.7838 
0.7966 

.88715 
.89423 
.90123 

1.2628 
1.2706 
1.2785 

.10134 
.10401 
.10670 

.61068 
.61691 
.62307 

0.74 
0.75 
0.76 

2.0959 
2.1170 
2.1383 

.32138 
.32572 
.33006 

.47711 
.47237 
.46767 

0.8094 
0.8223 
0.8353 

.90817 
.91504 
.92185 

1.2865 
1.2947 
1.3030  • 

.10942 
.11216 
.11493 

.62915 
.63515 
.64108 

0.77 
0.78 
0.79 

2.1598 
2.1815 
2.2034 

.33441 

..33875 
.34309 

.46301 
.45841 
.45384 

0.8484 
0.8615 
0.8748 

.92859 
.93527 
.94190 

1.3114 
1.3199 
1.3286 

.11773  [ 

.12055 

.12340 

.64693 
.65271 
.65841 

0.80 

2.2255 

..34744 

.44933 

0.8881 

.94846 

1.3374 

.12627 

.66404 

0.81 

0.82 
0.83 

2.2479 
2.2705 
2.2933 

..35178 
.35612 
.36046 

.44486 
.44043 
.43605 

0.9015 
0.9150 
0.9286 

.95498 
.96144 
.96784 

1.3464 
1.3555 
1.3647 

.12917 
.13209 
.13503 

.66959 
.67507 
.68048 

0.84 
0.85 
0.86 

2.3164 
2.3396 
2.3632 

.36481 
.36915 
.37349 

.43171 
.42741 
.42316 

0.9423 
0.9561 
0.9700 

.97420 
.98051 
.98677 

1.3740 
1.3835 
1.3932 

.13800 
.14099 
.14400 

.68581 
.69107 
.69626 

0.87 
0.88 
0.89 

2.3869 
2.4109 
2.4351 

.37784 
.38218 
.38652 

.41895 
.41478 
.41066 

0.9840 
0.9981 
1.0122 

.99299 
.99916 

.00528 

1.4029 
1.4128 
1.4229 

.14704 
.15009 
.15317 

.70137 
.70642 
.71139 

.71630 

0.90 

2.4596 

.39087 

.40657 

1.0265 

.01137 

1.4331 

.15627 

0.91 
0.92 
0.93 

2.4843 
2.5093 
2.5345 

.39521 
.39955 
.40389 

.40252 
.39852 
.39455 

1.0409 
1.0554 
1.0700 

.01741 
.02341 
.02937 

1.4434 
1.4539 
1.4645 

.15939 
.16254 
.16570 

.72113 
.72590 
.73059 

0.94 
0.95 
0.96 

2.5600 
2.5857 
2.6117 

.40824 
.41258 
.41692 

.39063 
.38674 
.38289 

1.0847 
1.0995 
1.1144 

.03530 
.01119 
.04704 

1.4753 

1.4862 
1.4973 

.16888 
.17208 
.17531 

.73522 

.73978 
.74428 

0.97 
0.98 
0.99 

2.6.379 
2.6645 
2.6912 

.42127 
.42561 
.42995 

.37908 
.37531 
.37158 

1.1294 
1.1446 
1.1598 

.05286 
.05864 
.06439 

1.5085 
1.5199 
1.5314 

1.5431 

.17855 
.18181 
.18509 

.18839 

.74870 
.75307 
.75736 

1.00 

2.7183 

.43429 

.36788 

1.1752 

.07011 

.76159 

118       Yalues  and  Logarithms  of  Hyperbolic  Functions 


00 

Value 

Logio 

Value 

Value   Logio 

CoshiK 

Value   Logio 

Tanha; 

Value 

1.00 

2.7183 

.43429 

.36788 

1.1752 

.07011 

1.5431 

.18839 

.76159 

1.01 
1.02 
1.03 

2.7456 
2.7732 
2.8011 

.43864 
.44298 
.44732 

.36422 
.36060 
.35701 

1.1907 
1.2063 
1.2220 

.07580 
.08146 
.08708 

1.5549 
1.5669 
1.5790 

.19171 
.19504 
.19839 

.76576 
.76987 
.77391 

1.04 
1.05 
1.06 

2.8292 
2.8577 
2.8864 

.45167 
.45601 
.46035 

.35345 
.34994 
.34646 

1.2379 
1.2539 
1.2700 

.09268 
.09825 
.10379 

1.5913 
1.6038 
1.6164 

.20176 
.20515 
.20855 

.77789 
.78181 
.78566 

1.07 
1.08 
1.09 

1.10 

2.9154 
2.9447 

2.9743 

.46470 
.46904 
.47338 

.34301 
.339(i0 
.33622 

1.2862 
1.3025 
1.3190 

.10930 
.11479 
.12025 

1.6292 
1.6421 
1.6552 

.21197 
.21541 

.21886 

.78946 
.79320 
.79688 

3.0042 

.47772 

.33287 

1.3356 

.12569 

1.6685 

.22233 

.80050 

1.11 
1.12 
1.13 

3.0344 
3.0649 
3.0957 

.48207 
.48641 
.49075 

.32956 
.32628 
.32303 

1.3524 
1.3693 
1.3863 

.13111 
.13649 
.14186 

1.6820 
1.6956 
1.7093 

.22582 
.22931 
.23283 

.80406 
.80757 
.81102 

1.14 
1.15 
1.16 

3.1268 
3.1582 
3.1899 

.49510 
.49944 
.50378 

.31982 
.31664 
.31349 

1.4035 
1.4208 
1.4382 

.14720 
.15253 
.15783 

1.7233 
1.7374 
1.7517 

.23636 
.23990 
.24346 

.81441 
.81775 
.82104 

1.17 
1.18 
1.19 

3.2220 
3.2544 
3.2871 

.50812 
.51247 
.51681 

.31037 
.30728 
.30422 

1.4558 
1.4735 
1.4914 

.16311 
.1683() 
.17360 

1.7662 
1.7808 
1.7957 

.24703 
.25062 
.25422 

.82427 
.82745 
.83058 

1.20 

3.3201 

.52115 

.30119 

1.5085 

.17882 

1.8107 

.25784 

.83365 

1.21 
1.22 
1.23 

3.3535 
3.3872 
3.4212 

.52550 
.52984 
.58418 

.29820 
.29523 
.29229 

1.5276 
1.5460 
1.5645 

.18402 
.18920 
.19437 

1.8258 
1.8412 
1.8568 

.26146 
.26510 
.26876 

.83668 
.83965 
.84258 

1.24 
1.25 
1.26 

3.4556 
3.4903 
3.5254 

.53853 
.54287 
.54721 

.28938 
.28650 
.28365 

1.5831 
1.6019 
1.6209 

.19951 
.20464 
.20975 

1.8725 
1.8884 
1.9045 

.27242 
.27610 
.27979 

.84546 
.84828 
.85106 

1.27 
1.28 
1.29 

3.5609 
3.5966 
3.6328 

.55155 
.55590 
.56024 

.28083 
.27804 
.27527 

1.6400 
1.6593 
1.6788 

.21485 
.21993 
.22499 

1.9208 
1.9373 
1.9540 

.28349 

.28721 
.29093 

.85380 
.85648 
.85913 

1.30 

3.6693 

.56458 

.27253 

1.6984 

.23004 

1.9709 

.29467 

.86172 

1.31 
1.32 
1.33 

3.7062 
3.7434 
3.7810 

.56893 
.57327 
.57761 

.26982 
.26714 
.26448 

1.7182 
1.7381 
1.7583 

.23507 
.24009 
.24509 

1.9880 
2.0053 
2.0228 

.29842 
.30217 
.30594 

.86428 
.86678 
.86925 

1.34 
1.35 
1.36 

3.8190 
3.8574 
3.8962 

.58195 
.58630 
.59064 

.26185 
.25924 
.25666 

1.7786 
1.7991 
1.8198 

.25008 
.25505 
.26002 

2.0404 
2.0583 
2.0764 

.30972 
.31352 
.31732 

.87167 
.87405 
.87639 

1.37 
1.38 
1.39 

3.9354 
3.9749 
4.0149 

.59498 
.59933 
.60367 

.25411 
.25158 
.24908 

1.8406 
1.8617 
1.8829 

.26496 
.26990 

.27482 

2.0947 
2.1132 
2.1320 

.32113 
.32495 

.32878 

.87869 
.88095 
.88317 

1.40 

4.0552 

.60801 

.24660 

1.9043 

.27974 

2.1509 

.33262 

.88535 

1.41 
1.42 
1.43 

4.0960 
4.1371 

4.1787 

.61236 
.61670 
.62104 

.24414 
.24171 
.23931 

1.9259 
1.9477 
1.9697 

.28464 
.28952 
.29440 

2.1700 
2.1894 
2.2090 

.33647 
.34033 
.34420 

.88749 
.88960 
.89167 

1.44 
1.45 
1.46 

4.2207 
4.2631 
4.3060 

.62538 
.62973 
.63407 

.23693 
.23457 
.23224 

1.9919 
2.0143 
2.0369 

.29926 
.30412 
.30896 

2.2288 
2.2488 
2.2691 

.34807 
.35196 
.35585 

.89370 
.89569 
.89765 

1.47 
1.48 
1.49 

4.3492 
4.3929 
4.4371 

.63841 
.64276 
.64710 

.22993 
.22764 
.22537 

2.0597 
2.0827 
2.1059 

.31379 
.31862 
.32343 

2.2896 
2.3103 
2.3312 

.35976 
.36367 
.36759 

.89958 
.90147 
.90332 

1.50 

4.4817 

.65144 

.22313 

2,1293 

.32823 

2.3524 

.37151 

.90515 

Values  and  Logarithms  of  Hyperbolic  Functions         119 


oc 

e 

Value 

Value 

Sinha? 

Value   Logio 

Cosho? 

Value   Logio 

TanhiT 

Value 

1.50 

4.4817 

.65144 

.22313 

2.1293 

.32823 

2.3524 

.37151 

.90515 

1.51 
1.52 
1.53 

4.5267 
4.5722 
4.6182 

.65578 
.66013 
.66447 

.22091 
.21871 
.21654 

2.1529 
2.1768 
2.2008 

.33303 
.33781 
.34258 

2.3738 
2.3955 
2.4174 

.37545 
.37939 
.38334 

.90694 
.90870 
.91042 

1.54 
1.55 
1.56 

4.6646 
4.7115 

4.7588 

.66881 
.67316 
.67750 

.21438 
.21225 
.21014 

2.2251 
2.2496 
2.2743 

.34735 
.35211 
.35686 

2.4395 
2.4619 
2.4845 

.38730 
.39126 
.39524 

.91212 
.91379 
.91542 

1.57 
1.58 
1.59 

4.8066 
4.8550 
4.9037 

.68184 
.68619 
.69053 

.20805 
.20598 
.20393 

2.2993 
2.3245 
2.3499 

.36160 
.36633 
.37105 

2.5073 
2.5305 
2.5538 

.39921 
.40320 
.40719 

.91703 
.91860 
.92015 

1.60 

4.9530 

.69487 

.20190 

2.3756 

.37577 

2.5775 

.41119 

.92167 

1.61 
1.62 
1.63 

5.0028 
5.0531 
5.1039 

.69921 
.70356 
.70790 

.19989 
.19790 
.19593 

2.4015 
2.4276 
2.4540 

.38048 
.38518 
.38987 

2.6013 
2.6255 
2.6499 

.41520 
.41921 
.42323 

.92316 
.92462 
.92606 

1.64 
1.65 
1.66 

5.1552 
5.2070 
5.2593 

.71224 
.71659 
.72093 

.19398 
.19205 
.19014 

2.4806 
2.5075 
2.5345 

.39456 
.39923 
.40391 

2.6746 
2.6995 
2.7247 

.42725 
.43129 
.43532 

.92747 
.92886 
.93022 

1.67 
1.68 
1.69 

5.3122 
5.3656 
5.4195 

.72527 
.72961 
.73396 

.18825 
.18637 
.18452 

2.5620 
2.5896 
2.6175 

.40857 
.41323 

.41788 

2.7502 
2.7760 
2.8020 

.43937 
.44341 
.44747 

.93155 
.93286 
.93415 

1.70 

5.4739 

.73830 

.18268 

2.6456 

.42253 

2.8283 

.45153 

.93541 

1.71 
1.72 
1.73 

5.5290 
5.5845 
5.6407 

.74264 
.74699 
.75133 

.18087 
.17907 

.17728 

2.6740 
2.7027 
2.7317 

.42717 
.43180 
.43643 

2.8549 
2.8818 
2.9090 

.45559 
.45966 
.46374 

.93665 
.93786 
.93906 

1.74 
1.75 
1.76 

5.6973 
5.7546 
5.8124 

.75567 
.76002 
.76436 

.17552 
.17377 
.17204 

2.7609 
2.7904 
2.8202 

.44105 
.44567 
.45028 

2.9364 
2.9642 
2.9922 

.46782 
.47191 
.47600 

.94023 
.94138 
.94250 

1.77 

1.78 
1.79 

5.8709 
5.9299 
5.9895 

.76870 
.77304 
.77739 

.17033 
.16864 
.16696 

2.8503 
2.8806 
2.9112 

.45488 
.45948 
.46408 

3.0206 
3.0492 

3.0782 

.48009 
.48419 
.48830 

.94361 
.94470 
.94576 

1.80 

6.0496 

.78173 

.16530 

2.9422 

.46867 

3.1075 

.49241 

.94681 

1.81 
1.82 
1.83 

6.1104 
6.1719 
6.2339 

.78(507 
.79042 
.79476 

.16365 
.16203 
.16041 

2.9734 
3.0049 
3.0367 

.47325 

.47783 
.48241 

3.1371 
3.1669 
3.1972 

.49652 
.50064 
.50476 

.94783 
.94884 
.94983 

1.84 
1.85 
1.86 

6.2965 
6.3598 
6.4237 

.79910 
.80344 
.80779 

.15882 
.15724 
.15567 

3.0689 
3.1013 
3.1340 

.48698 
.49154 
.49610 

3.2277 
3.2585 

3.2897 

.50889 
.51302 
.51716 

.9.5080 
.95175 
.95268 

1.87 
1.88 
1.89 

6.4883 
6.5535 
6.6194 

.81213 
.81647 
.82082 

.15412 
.15259 
.15107 

3.1671 
3.2005 
3.2341 

.50066 
.50521 
.50976 

3.3212 
3.3530 
3.3852 

.52130 
.52544 
.52959 

.95359 
.95449 
.95537 

1.90 

6.6859 

.82516 

.14957 

3.2682 

.51430 

3.4177 

.53374 

.95624 

1.91 
1.92 
1.93 

6.7531 
6.8210 
6.8895 

.82950 
.83385 
.83819 

.14808 
.14661 
.14515 

3.3025 
3.3372 
3.3722 

.51884 
.52338 
.52791 

3.4506 
3.4838 
3.5173 

.53789 
.54205 
.54621 

.95709 
.95792 
.95873 

1.94 
1.95 
1.96 

6.9588 
7.0287 
7.0993 

.84253 
.84687 
.85122 

.14370 
.14227 
.14086 

3.4075 
3.4432 
3.4792 

.53244 
.53696 
.54148 

3.5512 
3.5855 
3.6201 

.55038 
.55455 
.55872 

.95953 
.96032 
.96109 

1.97 
1.98 
1.99 

7.1707 
7.2427 
7.3155 

.85556 
.85990 
.86425 

.13946 
.13807 
.13670 

3.5156 
3.5523 
3.58M 

.54600 
.55051 
.55502 

3.6551 
3.6904 
3.7261 

.56290 
.56707 
.57126 

.96185 
.96259 
.96331 

2.00 

7.3891 

.86859 

.13534 

3.6269 

.55953 

3.7622 

.57544 

.96403 

120       Values  and  Logarithms  of  Hyperbolic  Functions 


nc 

e 

Value 

X 

Value 

Sinhi» 

Value   Logio 

Cosh  a? 

Value   Logio 

Tanhiz; 

Value 

200. 

7.3891 

.86859 

.13534 

3.6269 

.55953 

3.7622 

.57544 

.96403 

2.01 
2.02 
2.03 

7.4633 
7.5383 
7.6141 

.87293 

.87727 
.88162 

.13399 
.13266 
.13134 

3.6647 
3.7028 
3.7414 

.56403 
.56853 
.57303 

3.7987 
3.8355 
3.8727 

.57963 
.58382 
.58802 

.96473 
.96541 
.96609 

2.04 
2.05 
2.06 

7.6906 
7.7679 
7.8460 

.88596 
.89030 
.89465 

.13003 
.12873 
.12745 

3.7803 
3.8196 
3.8593 

.57753 
.58202 
.58650 

3.9103 
3.9483 
3.9867 

.59221 
.59641 
.60061 

.96675 
.96740 
.96803 

2.07 
2.08 
2.09 

7.9248 
80045 
8.0849 

.89899 
.90333 
.90768 

.12619 
.12493 
.12369 

3.8993 
3.9398 
3.9806 

.59099 
.59547 
.59995 

4.0255 
4.0647 
4.1043 

.60482 
.60903 
.61324 

.96865 
.96926 
.96986 

2.10 

8.1662 

.91202 

.12246 

4.0219 

.60443 

4.1443 

.61745 

.97045 

2.11 
2.12 
2.13 

8.2482 
8.3311 
8.4149 

.91636 
.92070 
.92505 

.12124 
.12003 
.11884 

4.0635 
4.1056 
4.1480 

.60890 
.61337 
.61784 

4.1847 
4.2256 
4.2669 

.62167 
.62589 
.63011 

.97103 
.97159 
.97215 

2.14 
2.15 
2.16 

8.4994 
8.5849 
8.6711 

.92939 
.93373 
.93808 

.11765 
.11648 
.11533 

4.1909 
4.2342 
4.2779 

.62231 
.62677 
.63123 

4.3085 
4.3507 
4.3932 

.63433 
.63856 
.64278 

.97269 
.97323 
.97375 

2.17 
2.18 
2.19 

8.7583 
8.8463 
8.9352 

.94242 
.94676 
.95110 

.11418 
.11304 
.11192 

4.3221 
4.3666 
4.4116 

.63569 
.64015 
.64460 

4.4362 
4.4797 
4.5236 

.64701 
.65125 
.65548 

.97426 
.97477 
.97526 

2.20 

9.0250 

.95545 

.11080 

4.4571 

.64905 

4.5679 

.65972 

.97574 

2.21 
2.22 
2.23 

9.1157 
9.2073 
9.2999 

.95979 
.96413 
.96848 

.10970 
.10861 
.10753 

4.5030 
4.5494 
4.5962 

.65350 
.65795 
.66240 

4.6127 
4.6580 
4.7037 

.66396 
.66820 
.67244 

.97622 
.97668 
.97714 

2.24 
2.25 
2.26 

9.3933 
9.4877 
9.5831 

.97282 
.97716 
.98151 

.10646 
.10540 
.10435 

4.6434 
4.6912 
4.7394 

.66684 
.67128 
.67572 

4.7499 
4.7966 
4.8437 

.67668 
.68093 
.68518 

.97759 
.97803 
.97846 

2.27 
2.28 
2.29 

9.6794 
9.7767 
9.8749 

.98585 
.99019 
.99453 

.10331 
.10228 
.10127 

4.7880 
4.8372 
4.8868 

.68016 
.68459 
.68903 

4.8914 
4.9395 
4.9881 

.68943 
.69368 
.69794 

.97888 
.97929 
.97970 

2.30 

9.9742 

.99888 

.10026 

4.9370 

.69346 

5.0372 

.70219 

.98010 

2.31 
2.32 
2.33 

10.074 
10.176 
10.278 

.00322 
.00756 
.01191 

.09926 
.09827 
.09730 

4.9876 
5.0387 
6.0903 

.69789 
.70232 
.70675 

5.0868 
5.1370 
5.1876 

.70645 
.71071 
.71497 

.98049 
.98087 
.98124 

2.34 
2.35 
2.36 

10.381 
10.486 
10.591 

.01625 
.02059 
.02493 

.09633 
.09537 
.09442 

5.1425 
5.1951 
5.2483 

.71117 
.71559 
.72002 

5.2388 
5.2905 
5.3427 

.71923 
.72349 

.72776 

.98161 
.98197 
.98233 

2.37 

2.38 
2.39 

10.697 
10.805 
10.913 

.02928 
.03362 
.03796 

.09348 
.09255 
.09163 

5.3020 
5.3562 
5.4109 

.72444 

.72885 
.73327 

5.3954 
5.4487 
5..W26 

.73203 
.73630 
.74056 

.98267 
.98301 
.98335 

2  40 

11.023 

04231 

.09072 

5.4662 

.73769 

5.5569 

.74484 

.98367 

2.41 
2.42 
2.43 

11.134 
11.246 
11.359 

.04665 
.05099 
.05534 

.08982 
.08892 
.08804 

5.5221 
5.5785 
5.6354 

.74210 
.74652 
.75093 

5.6119 
5.6674 
5.7235 

.74911 
.75338 
.75766 

.98400 
.98431 

.98462 

2.44 
2.45 
2.46 

11.473 
11.588 
11.705 

.05968 
.06402 
.06836 

.08716 
.08629 
.08543 

5.6929 
5.7510 
5.8097 

.75534 
.75975 
.76415 

5.7801 
5.8373 
5.8951 

.76194 
.76621 
.77049 

.98492 
.98522 
.98551 

2.47 
2.48 
2.49 

11.822 
11.941 
12.061 

.07271 
.07705 
.08139 

.08458 
.08374 
.08291 

5.8689 
5.9288 
5.9892 

.76856 
.77296 

.77737 

5.9535 
6.0125 
6.0721 

.77477 
.77906 
.78334 

.98579 
.98607 
.98635 

2.50 

12.182 

.08574 

.08208 

6.0502 

.78177 

6.1323 

.78762 

.98661 

Values  and  Logarithms  of  Hyperbolic  Functions       121 


0? 

Value   Logio 

Value 

Sinha? 

Value   Logio 

Cosh  a; 

Value   Logio 

Tanha? 

Value 

2.60 

12.182 

.08574 

.08208 

6.0502 

.78177 

6.1323 

.78762 

.98661 

2.51 
2.52 
2.53 

12.305 
12.429 
12.554 

.09008 
.09442 
.09877 

.08127 
.08046 
.07966 

61118 
6.1741 
6.2369 

.78617 
.79057 
.79497 

6.1931 
6.2545 
6.3166 

.79191 
.79619 
.80048 

.98688 
.98714 
.98739 

2.54 
2.55 
2.56 

12.680 
12.807 
12.936 

.10311 
.10745 
.11179 

.07887 
.07808 
.07730 

6.3004 
6.3645 
6.4293 

.79937 
.80377 
.80816 

6.3793 
6.4426 
6.5066 

.80477 
.80906 
.81335 

.98764 
.98788 
.98812 

2.57 
2.58 
2.59 

13.0(J6 
13.197 
13.330 

.11614 
.12048 
.12482 

.07654 

.07577 
.07502 

6.4946 
6.5607 
6.6274 

6.6947 

6.7628 
6.8315 
6.9008 

.81256 
.81695 
.82134 

.82573 

.83012 
.83451 
.83890 

6.5712 
6.6365 
6.7024 

.81764 
.82194 
.82623 

.98835 
.98858 
.98881 

2.60 

2.61 
2.62 
2.63 

13.464 

.12917 

.07427 

6.7690 

6.8363 
6.9043 
6.9729 

.83052 

.83482 
.83912 
.84341 

.98903 

13.599 
13.736 
13.874 

.13351 
.13785 
.14219 

.07353 
.07280 
.07208 

.98924 
.98946 
.98966 

2.64 
2.65 
2.66 

14.013 
14.154 
14.296 

.14654 
.15088 
.15522 

.07136 
.07065 
.06995 

6.9709 
7.0417 
7.1132 

.84329 
.84768 
.85206 

7.0423 
7.1123 
7.1831 

.84771 
.85201 
.85631 

.98987 
.99007 
.99026 

2.67 
2.68 
2.69 

2.70 

2.71 

2.72 
2*73 

14.440 
14.585 
14.732 

.15957 
.16391 
.16825 

.06925 
.06856 
.06788 

7.1854 
7.2583 
7.3319 

.85645 
.86083 
.86522 

7.2546 

7.3268 
7.3998 

.86061 
.86492 
.86922 

.99045 
.99064 
.99083 

14.880 

.17260 

.06721 

7.4063 

•  7.4814 
7.5572 
7.6338 

.86960 

.87398 
.87836 
.88274 

7.4735 

7.5479 
7.6231 
7.6991 

.87352 

.87783 
.88213 
.88644 

.99101 

.99118 
.99136 
.99153 

15.029 
15.180 
15.333 

.17694 
.18128 
.18562 

.06654 
.06587 
.06522 

2.74 
2.75 
2.76 

15.487 
15.643 
15.800 

.18997 
.19431 

.19865 

.06457 
.06393 
.06329 

7.7112 
7.7894 
7.8683 

.88712 
.89150 
.89588 

7.7758 
7.8533 
7.9316 

.89074 
.89505 
.89936 

.99170 
.99186 

.99202 

2.77 
2.78 
2.79 

2.80 

15.959 
16.119 
16.281 

.20300 
.20734 
.21168 

.06266 
.06204 
.06142 

7.9480 
8.0285 
8.1098 

.90026 
.90463 
.90^)01 

8.0106 
8.0905 
8.1712 

8.2527 

.90367 
.90798 
.91229 

.91660 

.99218 
.99233 
.99248 

16.445 

.21602 

.06081 

8.1919 

.91339 

.99263 

2.81 
2.82 
2.83 

16.610 
16.777 
16.945 

.22037 
.22471 
.22905 

.06020 
.05961 
.05901 

8.2749 
8.3586 
8.4432 

.91776 
.92213 
.92651 

8.3351 
8.4182 
8.5022 

.92091 
.92522 
.92953 

.99278 
.99292 
.99306 

2.84 

2.85 
2.86 

17.116 

17.288 
17.462 

.23340 
.23774 
.24208 

.05843 
.05784 
.05727 

8.5287 
8.6150 
8.7021 

.91^88 
.93525 
.93963 

8.5871 
8.6728 
8.7594 

.93385 
.93816 
.94247 

.99320 
.99333 
.99346 

2.87 
2.88 
2.89 

2.90 

17.637 
17.814 
17.993 

.24643 
.25077 
.25511 

.05670 
.05613 
.05558 

8.7902 
8.8791 
8.9689 

9.0596 

.94400 
.94837 
.95274 

.95711 

8.8469 
8.9352 
9.0244 

.94679 
.95110 
.95542 

.99359 

.99372 
.99384 

18.174 

.25945 

.05502 

9.1146 

.95974 

.^)9396 

2.91 
2.92 
2.93 

18.357 
18.541 

18.728 

.26380 
.26814 

.27248 

.05448 
.05393 
.05340 

9.1512 
9.2437 
9.3371 

.96148 
.96584 
.97021 

9.2056 
9.2976 
9.3905 

.96405 
.96837 
,97269 

.99408 
.99420 
.99431 

2.94 
2  95 
2.96 

18.916 
19.106 
19.298 

.27683 
.28117 
.28551 

.05287 
.05234 
.05182 

9.4315 
9.5268 
9.6231 

.97458 
.97895 
.98331 

9.4844 
9.5791 
9.6749 

.97701 
.98133 
.98565 

.99443 

.99454 

.  .99464 

2.97 
2.98 
2.99 

19.492 
19.688 
19.886 

.28985 
.29420 
.29854 

.05130 
.05079 
.05029 

9.7203 
9.8185 
9.9177 

.98768 
.99205 
.99641 

9.7716 
9.8693 
9.9680 

.98997 
.99429 
.99861 

.99475 
.99485 
.99496 

3.00 

20.086 

.30288 

.04979 

10.018 

.00078 

10.068 

.00293 

.99505 

122       Values  and  Logarithms  of  Hyperbolic  Functions 


iC 

e 

Value 

Value 

SinhiT 

Value   Logio 

Cosh  i» 
Value   Logio 

Tanh  oc 

A'alue 

3.00 

20.086 

.30288 

.04979 

10.018 

.00078 

10.068 

.00293 

.99505 

3.05 
3.10 
3.15 

21.115 
22.198 
23.336 

.32460 
.34631 
.36803 

.04736 
.04505 
.04285 

10.534 
11.076 
11.646 

.02259 
.04440 
.06619 

10.581 
11.122 
11.690 

.02454 
.04616 
.06780 

.99552 
.99595 
.99631 

3.20 
3.25 
3.30 

24.533 
25.790 
27.113 

.38974 
.41146 
.43317 

.04076 

.03877 
.03688 

12.246 
12.876 
13.538 

.08799 
.10977 
.13155 

12.287 
12.915 
13.575 

.08943 
.11108 
.13273 

.99668 
.99700 
.99728 

3.35 
3.40 
3.45 

28.503 
29.964 
31.500 

.45489 
.47660 
.49832 

.03508 
.03.337 
.03175 

14.234 
14.965 
15.734 

.15332 
.17509 
.19685 

14.269 
14.999 
15.766 

.15439 
.17605 
.19772 

.99754 
.99777 
.99799 

3.50 

33.115 

.52003 

.03020 

16.543 

.21860 

16.573 

.21940 

.99818 

3.55 
3.60 
3.65 

34.813 
;i6.598 
38.475 

.54175 
.56346 
.58517 

.02872 
.02732 
.02599 

17.392 
18.286 
19.224 

.24036 
.26211 
.28385 

17.421 
18.313 
19.250 

.24107 
.26275 
.28444 

,99833 
.99851 
.99865 

3  70 
3.75 
3.80 

40.447 
42.521 
44.701 

.60689 
.62860 
.65032 

.02472 
.02352 
.92237 

20.211 
21.249 
22.339 

.30559 
.32733 
.34907 

20.236 
21.272 
22.362 

.30612 
.32781 
.34951 

.99878 
.99889 
.99900 

3.85 
3.90 
3.95 

46.993 
49.402 
51.935 

.67203 
.69375 
.71546 

.02128 
.02024 
.01925 

23.486 
24.691 
25.958 

.37081 
.39254 
.41427 

23.507 
24.711 
25.977 

.37120 
.39290 
.41459 

.99909 
99918 
.99926 

4.00 

4.10 
4.20 
4.30 

54.598 

.73718 

.01832 

27.290 

.43600 

27.308 

.43()29 

.99933 

(50.340 
6().()86 
73.700 

.780(>1 
.82404 
.86747 

.01(]57 
.01500 
.01357 

30.162 
33.33() 
36.843 

.47946 
.52291 
.56636 

30.178 
S3.. Sol 
:3(>.857 

.47970 
.52310 
.56652 

.99945 
.99955 
.99963 

4.40 
4.50 
4.60 

81.451 
90.017 
99.484 

.910^)0 
.95433 
.99775 

.01227 
.01111 
.01005 

40.719 
45.003 
49.737 

.60980 
.65324 
.6J)668 

40.732 
45.014 
49.747 

.60993 
.65335 
.69677 

.9^)970 
.99^)75 
.99980 

4.70 
4.80 
4.90 

109.95 
121.61 
134.29 

.04118 
.08461 
.12804 

.00910 
.00823 
.00745 

54.9(i9 
()0.751 
67.141 

.74012 
.78355 
.82()99 

54.978 
60.759 
67.149 

.74019 
.78361 
.82704 

.99^)83 
.99986 
.99989 

5.00 

148.41 

.17147 

.00674 

74.203 

.87042 

74.210 

.87046 

.99991 

5.10 
5.20 
5.30 

164.02 
181.27 
200.34 

.214^)0 
.25833 
.30176 

.00610 
.00552 
.00199 

82.011 
90.633 
100.17 

.91386 
.95729 
.00074 

82.014 
90.(i39 
100.17 

.91389 
.95731 
.00074 

.99993 
.99994 
.99995 

5.40 
5.50 
5.60 

221.41 
244.69 
270.43 

.34519 
.38862 
.43205 

.00452 
.00409 
.00370 

110.70 
122.1U 
135.21 

.04415 
.08768 
.13101 

110.71 
122..T) 
135.22 

.04417 
.087(30 
.13103 

.99996 
.99997 
.99997 

5.70 
5.80 
5.t)0 

298.87 
330.:30 
3()5.04 

.47548 
.51891 
.56234 

.00335 
.00303 
.00274 

149.43 
165.15 
182.52 

.17444 
.21787 
.26130 

149.44 
1(J5.15 
182.52 

.17445 

.21788 
.26131 

.99998 
.99998 
.99998 

6.00 

403.43 

.60577 

.00248 

.00193 
.00150 
.00117 

201.71 

.30473 

201.72 

.30474 

.9^)999 

6.25 
6.50 
6.75 

518.01 
665.14 
854.06 

.71434 
.82291 
.93149 

259.01 
332.57 
427.03 

.41331 

.52188 
.63046 

259.01 
332.57 
427.03 

.41331 
.52189 
.63046 

.99999 
1.0000 
1.0000 

7.00 
7.50 
8.00 

1096.6 

1808.0 
2981.0 

.04006 
.25721 
.47436 

.00091 
.00055 
.00034 

548.32 
904.02 
1490.5 

.73903 
.95618 
.17333 

548.32 
904.02 
1490.5 

.73903 
.95618 
.17333 

1.0000 
1.0000 
1.0000 

8.50 
9.00 
9.50 

4914.8 
8103.1 
133(30. 

.69150 
.90865 
,12580 

.00020 
.00012 
.00007 

2457.4 
4051.5 
6679.9 

.39047 
.60762 
.82477 

2457.4 
4051.5 
6679.9 

.39047 
.60762 
.82477 

1.0000 
1.0000 
1.0000 

10.00 

22026. 

.34294 

.00005 

11013. 

.04191 

11013. 

.04191 

1.0000 

Table  X  —  Values  and  Logarithms  of  Haversines       123 


[Characteristics  of  Logai 

ithnis  omitted  - 

-  determine  by 

rule  from  the  value] 

o 

0 

10' 

20' 

30' 

40' 

50' 

Value 

Logio 

Value 

Logio 

Value 

I-ogio 

Value 

Log.o 

Value 

Log,o 

Value  Logio 

~o" 

.0000 

.0000  4.3254 

.0000  4.9275 

.0000  5.2796 

.0000 

5.5295 

.0001  5.7223 

1 

.0001  5.8817 

.0001  6.0156 

.0001  6.1315 

.0002 

.2338 

.0002 

.3254 

.0003 

.4081 

2 

.0003 

.4837 

.0004 

.5532 

.0004 

.6176 

.0005 

.6775 

.0005 

.7336 

.0006 

.7862 

3 

.0007 

.8358 

.0008 

.8828 

.0008 

.9273 

.0009 

.9697 

.0010 

.0101 

.0011 

,0487 

4 

.0012 

.0856 

.0013 

.1211 

.0014 

.1551 

.0015 

.1879 

.0017 

.2195 

.0018 

.2499 

5 

.0019 

.2793 

.0020 

.3078 

.0022 

.3354 

.0023 

.3621 

.0024 

.3880 

.0026 

.4132 

6 

.0027 

.4376 

.0029 

.4614 

.0031 

.4845 

.0032 

.5071 

.0034 

.5290 

.0036 

.5^504 

7 

.0037 

.5713 

.0039 

.5918 

.0041 

.6117 

.0043 

.6312 

.0045 

.6503 

.0047 

.6689 

8 

.0049 

.6872 

.0051 

.7051 

.0053 

.7226 

.0055 

.7397 

.0057 

.7566 

.0059 

.7731 

9 

.0062 

.7893 

.0064 

.8052 

.0066 

.8208 

.0069 

.83(51 

.0071 

.8512 

.0073 

.8660 

10 

.0076 

.8806 

.0079 

.8949 

.0081 

.9090 

.0084 

.9229 

.0086 

.9365 

.0089 

.9499 

11 

.0092 

.9631 

.0095 

.9762 

.0097 

.98iX) 

.0100 

.0016 

.0103 

.0141 

.010(5 

.0264 

12 

.0109 

.0385 

.0112 

.0504 

.0115 

.0622 

.0119 

.0738 

.0122 

.0853 

.0125 

.096(5 

13 

.0128 

.1077 

.0131 

.1187 

.0135 

.1296 

.0138 

.1404 

.0142 

.1510 

.0145 

.1614 

14 

.0149 

.1718 

.0152 

.1820 

.0156 

.1921 

.0159 

.2021 

.0163 

.2120 

.0167 

.2218 

15 

.0170 

.2314 

.0174 

.2409 

.0178 

.2504 

.0182 

.2597 

.0186 

.2689 

.0190 

.2781 

k; 

.0194 

.2871 

.0198 

.2961 

.0202 

.3049 

.0206 

.3137 

.0210 

.3223 

.0214 

.3309 

17 

.0218 

.3394 

.0223 

.3478 

.0227 

.35(51 

.0231 

.3644 

.0236 

.3726 

.0240 

.380() 

18 

.0245 

.3887 

.0249 

.3966 

.0254 

.4045 

.0258 

.4123 

.0263 

.4200 

.0268 

.4276 

19 

.0272 

.4352 

.0277 

.4427 

.0282 

.4502 

.0287 

.4576 

.0292 

.4649 

.0297 

.4721 

20 

.0:^2 

.4793 

.0307 

.4865 

.0312 

.4936 

.0317 

.5006 

.0322 

.5075 

.0327 

.5144 

21 

.0332 

.5213 

.0337 

.5281 

.0.343 

.5348 

.0348 

.5415 

.0353 

.5481 

.0359 

.5547 

22 

.0:364 

.5612 

.0370 

.5677 

.0375 

.5741 

.0381 

.5805 

.0386 

.5868 

.0392 

.5931 

23 

.0397 

.5993 

.0403 

.6055 

.0409 

.6116 

.0415 

.6177 

.0421 

.(5238 

.0426 

.6298 

24 

.0432 

.6357 

.0438 

.6417 

.0444 

.6476 

.0450 

.6534 

.0456 

.6592 

.0462 

.6650 

25 

.0468 

.6707 

.0475 

.6764 

.0481 

.6820 

.0487 

.()876 

.0493 

.6932 

.0500 

.()987 

2lj 

.0506 

.7042 

.0512 

.7096 

.0519 

.7151 

.0525 

.7204 

.0532 

.7258 

.0538 

.7311 

27 

.0545 

.73(^4 

.0552 

.7416 

.0.558 

.7468 

.0565 

.7520 

.0572 

.7572 

.0578 

.762:5 

28 

.0585 

.7673 

.0592 

.7724 

.0599 

.7774 

.0(306 

.7824 

.0613 

.7874 

.0()20 

.7923 

29 

.0627 

.7972 

.0634 

.8020 

.0641 

.8069 

.0()48 

.8117 

.0655 

.8165 

.0(363 

.8213 

30 

.0670 

.8260 

.0()77 

.8307 

.0684 

.8354 

.0(592 

.8400 

.0699 

.8446 

.0707 

.8492 

31 

.0714 

.8538 

.0722 

.a^83 

.0729 

.8629 

.0737 

.8673 

.0744 

.8718 

.0752 

.8763 

32 

.0760 

.8807 

.0767 

.8851 

.0775 

.8894 

.0783 

.8938 

.0791 

.8981 

.0799 

.9024 

33 

.0807 

.^67 

.0815 

.9109 

.0823 

.9152 

.0831 

.9194 

.0839 

.9236 

.0847 

.9277 

34 

.0855 

.9319 

.0863 

.93(X) 

.0871 

.9401 

.0879 

.9442 

.0888 

.9482 

.0896 

.9523 

35 

.0i)04 

.9563 

.0913 

.9603 

.0921 

.9643 

.0929 

.9682 

.0938 

.9722 

.0946 

.9761 

36 

.0955 

.9800 

.0963 

.9838 

.0972 

.9877 

.0981 

.9915 

.0989 

.9954 

.0998 

.9992 

37 

.1007 

.0030 

.1016 

.00<)7 

.1024 

.0105 

.1033 

.0142 

.1042 

.0179 

.1051 

.0216 

38 

.1060 

.0253 

.10(;9 

.0289 

.1078 

.0326 

.1087 

.03(52 

.1096 

.0398 

.1105 

.0434 

39 

.1114 

.0470 

.1123 

.0505 

.1133 

.0541 

.1142 

.0576 

.1151 

.0611 

.1160 

.0646 

40 

.1170 

.0681 

.1179 

.0716 

.1189 

.0750 

.1198 

.0784 

.1207 

.0817 

.1217 

.0853 

41 

.1226 

.0887 

.1236 

.0920 

.1246 

.0954 

.1255 

.0987 

.1265 

.1021 

.1275 

.1054 

42 

.1284 

.1087 

.1294 

.1119 

.1304 

.1152 

.1314 

.1185 

.1323 

.1217 

.1333 

.1249 

43 

.11^3 

.1282 

.1353 

.1314 

.1363 

.1345 

.1373 

.1377 

.1383 

.1409 

.1393 

.1440 

44 

.1403 

.1472 

.1413 

.1503 

.1424 

.1534 

.1434 

.1565 

.1444 

.1596 

.1454 

.1626 

45 

.1464 

.1657 

.1475 

.1687 

.1485 

.1718 

.1495 

.1748 

.1506 

.1778 

.1516 

.1808 

46 

.1527 

.1838 

.1538 

.1867 

.1548 

.1897 

.1558 

.1926 

.1569 

.1956 

.1579 

.1985 

47 

.1590 

.2014 

.1600 

.2043 

.1611 

.2072 

.1622 

.2101 

.1633 

.2129 

.1644 

.2158 

48 

.1(>54 

.2186 

.1665 

.2215 

.1676 

.2243 

.1687 

.2271 

.1698 

.2299 

.1709 

.2327 

49 

.1720 

.2355 

.1731 

.2382 

.1742 

.2410 

.1753 

.2437 

.1764 

.2465 

.1775 

.2492 

50 

.1786 

.2519 

.1797 

.2546 

.1808 

.2573 

.1820 

.2600 

.1831 

.2627 

.1842 

.2653 

51 

.1853 

.2680 

.1865 

.2706 

.1876 

.2732 

.1887 

.2759 

.1899 

.2785 

.1910 

.2811 

52 

.1922 

.2837 

.1933 

.2863 

.1945 

.2888 

.1956 

.2914 

.1968 

.2940 

.1979 

.2965 

53 

.1991 

.2991 

.2003 

.3016 

.2014 

.3041 

.2026 

.3066 

.2038 

.3091 

.2049 

.3116 

54 

.2061 

.3141 

.2073 

.3166 

.2085 

.3190 

.2096 

.3215 

.2108 

.3239 

.2120 

.3264 

55 

.2132 

.3288 

.2144 

.3312 

.2156 

.3336 

.2168 

.3361 

.2180 

.3384 

.2192 

.3408 

56 

.2204 

.3432 

.2216 

.3456 

.2228 

.3480 

.2240 

.3503 

.2252 

.3527 

.2265 

.3550 

57 

.2277 

.3573 

.2289 

.3596 

.2301 

.3620 

.2314 

.3643 

.2326 

.3666 

.2338 

.3689 

58 

.2350 

.3711 

.2363 

.3734 

.2375 

.3757 

.2388 

.3779 

.2400 

.3802 

.2412 

.3824 

59 

.2425 

.3847 

.2437 

.3869 

.2450 

.3891 

.2462 

.3913 

.2475 

.3935 

.2487 

3957 

124  Values  and  Logarithms  of  Haversines 

[Characteristics  of  Logarithms  omitted  —  determine  by  rule  from  the  value] 


[X 


• 

0' 

10' 

20' 

30' 

40' 

50' 

Value 

Logio 

Value 

Logio 

Value 

Logio 

Value  Logio 

Value 

Logio 

Value  Logio 

60 

.2500 

.3979 

.2513 

.4001 

.2525 

.4023 

.2538 

.4045 

.2551 

.4006 

.2563  .4088 

61 

.2576 

.4109 

.2589 

.4131 

.2601 

.4152 

.2614 

.4173 

.2627 

.4195 

.2640  .4216 

62 

.2653 

.4237 

.2665 

.4258 

.2678 

.4279 

.2691 

.4300 

.2704 

.4320 

.2717  .4341 

63 

.2730 

.4362 

.2743 

.4382 

.2756 

.4403 

.2769 

.4423 

.2782 

.4444 

.2795  .4464 

64 

.2808 

.4484 

.2821 

.4504 

.2834 

.4524 

.2847 

.4545 

.2861 

.4565 

.2874  .4584 

65 

.2887 

.4604 

.2900 

.4624 

.2913 

.4644 

.2927 

.4664 

.2940 

.4683 

.2953  .4703 

66 

.2966 

.4722 

.2980 

.4742 

.2993 

.4761 

.3006 

.4780 

.3020 

.4799 

.3033  .4819 

67 

.3046 

.4838 

.3060 

.4857 

.3073 

.4876 

.3087 

.4895 

.3100 

.4914 

.3113  .4932 

68 

.3127 

.4951 

.3140 

.4970 

.3154 

.4989 

.3167 

.5007 

.3181 

.5026 

.3195  .6044 

69 

.3208 

.5063 

.3222 

.5081 

.3235 

.5099 

.3249 

.5117 

.3263 

.5136 

.3276  .5154 

70 

.3290 

.5172 

.3304 

.5190 

.3317 

.5208 

.3331 

.5226 

.3345 

.5244 

.3358  .5261 

71 

.3372 

.5279 

.3386 

.5297 

.3400 

.5314 

.3413 

.5332 

.3427 

.5349 

.3441  .5367 

72 

.3455 

.5384 

.34()9 

.5402 

.3483 

.5419 

.3496 

.5436 

.3510 

.5454 

.3524  .5471 

73 

.3538 

.5488 

.3552 

.5505 

.3566 

.5522 

.3580 

.5539 

.3594 

.5556 

.3608  .5572 

74 

.3622 

.5589 

.3636 

.5606 

.3650 

.5623 

.3664 

.5639 

.3678 

.5656 

.3692  .5672 

75 

.3706 

.5689 

.3720 

.5705 

.3734 

.5722 

.3748 

.5738 

.3762 

.5754 

.3776  .5771 

76 

.3790 

.5787 

.3805 

.5803 

.3819 

.5819 

.3833 

.5835 

.3847 

.5851 

.3861  .5867 

77 

.3875 

.5883 

.3889 

.5899 

.3904 

.5915 

.3918 

.5930 

.3932 

.5946 

.3946  .5962 

78 

.3960 

.5977 

.3975 

.5993 

.3989 

.6009 

.4003 

.6024 

.4017 

.6039 

.4032  .6055 

79 

.4046 

.6070 

.4060 

.6085 

.4075 

.6101 

.4089 

.6116 

.4103 

.6131 

.4117  .6146 

80 

.4132 

.6161 

.4146 

.6176 

.4160 

.6191 

.4175 

.6206 

.4189 

.6221 

.4203  .6236 

81 

.4218 

.6251 

.4232 

.6266 

.4247 

.6280 

.4261 

.6295 

.4275 

.6310 

.4290  .6324 

82 

.4304 

.6339 

.4319 

.(5353 

.4333 

.6368 

.4347 

.6382 

.4362 

.6397 

.4376  .6411 

83 

.4391 

.6425 

.4405 

.6440 

.4420 

.6454 

.4434 

.6468 

.4448 

.6482 

.4463  .6496 

84 

.4477 

.6510 

.4492 

.6524 

.4506 

.6538 

.4521 

.6552 

.4535 

.6566 

.4550  .6580 

85 

.4564 

.6594 

.4579 

.6607 

.4593 

.6621 

.4608 

.6635 

.4622 

.6649 

.4637  .6662 

86 

.4651 

.6676 

.4666 

.6689 

.4680 

.6703 

.4695 

.6716 

.4709 

.6730 

.4724  .6743 

87 

.4738 

.6756 

.4753 

.6770 

.4767 

.6783 

.4782 

.6796 

.4796 

.6809 

.4811  .6822 

88 

.4826 

.6835 

.4840 

.6848 

.4855 

.6862 

.4869 

.6875 

.4884 

.6887 

.4898  .6900 

89 

.4913 

.6913 

.4937 

.6926 

.4942 

.6939 

.4956 

.6952 

.4971 

.6964 

.4985  .6977 

90 

.5000 

.6990 

.5015 

.7002 

.5029 

.7015 

.5044 

.7027 

.5058 

.7040 

.5073  .7052 

91 

.5087 

.7065 

.5102 

.7077 

.5116 

.7090 

.5131 

.7102 

.5145 

.7114 

.5160  .7126 

92 

.5174 

.7139 

.5189 

.7151 

.5204 

.7163 

.5218 

.7175 

.5233 

.7187 

.5247  .7199 

93 

.5262 

.7211 

.5276 

.7223 

.5291 

.72.35 

.5305 

.7247 

.5320 

.7259 

.5334  .7271 

94 

.5349 

.7283 

.5363 

.7294 

.5378 

.7306 

.5392 

.7318 

.5407 

.7329 

.5421  .7341 

95 

.5436 

.7353 

.5450 

.7364 

.5465 

.7376 

.5479 

.7387 

.5494 

.7399 

.5508  .7410 

96 

.5523 

.7421 

.5537 

.7433 

.5552 

.7444 

.5566 

.7455 

.5580 

.7467 

.5595  .7478 

97 

.5609 

.7489 

.5624 

.7500 

.5638 

.7511 

.5653 

.7523 

.5667 

.7534 

.5082  .7545 

98 

.5696 

.7556 

.5710 

.7567 

.5725 

.7577 

.5739 

.7588 

.5753 

.7599 

.5768  .7610 

99 

.5782 

.7621 

.5797 

.7632 

.5811 

.7642 

.5825 

.7653 

.5840 

.7664 

.5854  .7674 

100 

.5868 

.7685 

.5883 

.7696 

.5897 

.7706 

.5911 

.7717 

.5925 

.7727 

.5940  .7738 

101 

.5954 

.7748 

.5968 

.7759 

.5983 

.7769 

.5997 

.7779 

.6011 

.7790 

.6025  .7800 

102 

.6040 

.7810 

.6054 

.7820 

.6068 

.7830 

.6082 

.7841 

.6096 

.7851 

.6111  .7861 

103 

.6125 

.7871 

.6139 

.7881 

.6153 

.7891 

.6167 

.imi 

.6181 

.7911 

.6195  .7921 

104 

.6210 

.7931 

.6224 

.7940 

.6238 

.7950 

.6252 

.7960 

.6266 

.7970 

.6280  .7980 

105 

.6294 

.7989 

.6308 

.7999 

.6322 

.8009 

.6336 

.8018 

.6350 

.8028 

.6364  .8037 

106 

.6378 

.8047 

.6392 

.8056 

.6406 

.8066 

.6420 

.8075 

.6434 

.8085 

.6448  .8094 

107 

.6462 

.8104 

.6476 

.8113 

.6490 

.8122 

.6504 

.8131 

.6517 

.8141 

.6531  .8150 

108 

.6545 

.8159 

.6559 

.8168 

.6573 

.8177 

.6587 

.8187 

.6600 

.8196 

.6614  .8205 

109 

.6628 

.8214 

.6642 

.8223 

.6655 

.8232 

.6669 

.8241 

.6683 

.8250 

.6696  .8258 

110 

.6710 

.8267 

.6724 

.8276 

.6737 

.8285 

.6751 

.8294 

.6765 

.8302 

.6778  .8311 

111 

.6792 

.8320 

.6805 

.8329 

.6819 

.8337 

.6833 

.8346 

.6846 

.8354 

.6860  .8363 

112 

.6873 

.8371 

.6887 

.8380 

.6900 

.8388 

.6913 

.8397 

.6927 

.8405 

.6940  .8414 

113 

.6954 

.8422 

.6967 

.8430 

.6980 

.8439 

.6994 

.8447 

.7007 

.8455 

.7020  .8464 

114 

.7034 

.8472 

.7047 

.8480 

.7060 

.8488 

.7073 

.8496 

.7087 

.8504 

.7100  .8513 

116 

.7113 

.8521 

.7126 

.8529 

.7139 

.8537 

.7153 

.8545 

.7166 

.8553 

.7179  .8561 

116 

.7192 

.8568 

.7205 

.8576 

.7218 

.8584 

.7231 

.8592 

.7244 

.8600 

.7257  .8608 

117 

.7270 

.8615 

.7283 

.8623 

.7296 

.8631 

.7309 

.8638 

.7392 

.8646 

.7335  .8654 

118 

.7347 

.8661 

.7360 

.8669 

.7373 

.8676 

.7386 

.8684 

.7399 

.8691 

.7411  .8699 

119 

.7424 

.8706 

.7437 

.8714 

.7449 

.8721 

.7462 

.8729 

.7475 

.8736 

.7487  .8743 

X]  Values  and  Logarithms  of  Haversines 

[Characteristics  of  Logarithms  omitted  —  determine  by  rule  from  the  value] 


125 


o 

0 

f 

10' 

20' 

30' 

40' 

50' 

Value 

Logio 

Value 

Logio 

Value  Logio 

Value 

Logio 

Value  Logjo 

Value  Logxo 

120 

.7500 

.8751 

.7513 

.8758 

.7525 

.8765 

.7538 

.8772 

.7550 

.8780 

.7563  .8787 

121 

.7575 

.8794 

.7588 

.8801 

.7600 

.8808 

.7612 

.8815 

.7625 

.8822 

.7637  .8829 

122 

.7650 

.8836 

.7662 

.8843 

.7674 

.8850 

.7686 

.8857 

.7699 

.8864 

.7711  .8871 

123 

.7723 

.8878 

.7735 

.8885 

.7748 

.8892 

.7760 

.8898 

.7772 

.8905 

.7784  .8912 

124 

.7796 

.8919 

.7808 

.8925 

.7820 

.8932 

.7832 

.8939 

.7844 

.8945 

.7856  .8952 

125 

.7868 

.8959 

.7880 

.8965 

.7892 

.8972 

.7904 

.8978 

.7915 

.8985 

.7927  .8991 

12() 

.7939 

.8998 

.7951 

.9004 

.7962 

.9010 

.7974 

.9017 

.7986 

.9023 

.7997  .9030 

127 

.8009 

.9036 

.8021 

.9042 

.8032 

.9048 

.8044 

.9055 

.8055 

.9061 

.8067  .9067 

128 

.8078 

.9073 

.8090 

.9079 

.8101 

.9085 

.8113 

.9092 

.8124 

.9098 

.8135  .9104 

129 

.8147 

.9110 

.8158 

.9116 

.8169 

.9122 

.8180 

.9128 

.8192 

.9134 

.8203  .9140 

130 

.8214 

.9146 

.8225 

.9151 

.8236 

.9157 

.8247 

.9163 

.8258 

.9169 

.8269  .9175 

i:n 

.8280 

.9180 

.8291 

.9186 

.8302 

.9192 

.8313 

.9198 

.8324 

.9203 

.8335  .9209 

132 

.8346 

.9215 

.8356 

.9220 

.8367 

.9226 

.8378 

.9231 

.8389 

.9237 

.8399  .9242 

133 

.8410 

.9248 

.8421 

.9253 

.8431 

.9259 

.8442 

.9264 

.8452 

.9270 

.8463  .9275 

134 

.8473 

.9281 

.8484 

.9286 

.8494 

.9291 

..8501 

.9297 

.8515 

.9302 

.8525  .9307 

135 

.8536 

.9312 

.8546 

.9318 

.8556 

.9323 

.8566 

.9328 

.8576 

.9333 

.8587  .9338 

136 

.8597 

.9343 

.8607 

.9348 

.8617 

.9353 

.8627 

.9359 

.8637 

.9364 

.8647  .9369 

137 

.8657 

.9374 

.8667 

.9379 

.8677 

.9383 

.8686 

.9388 

.8696 

.9393 

.8706  .9398 

138 

.8716 

.9403 

.8725 

.9408 

.8735 

.9413 

.8745 

.9417 

.8754 

.9422 

.8764  .9427 

139 

.8774 

.9432 

.8783 

.9436 

.8793 

.9441 

.8802 

.9446 

.8811 

.9450 

.8821  .9455 

140 

.8830 

.9460 

.8840 

.9464 

.8849 

.9469 

.8858 

.9473 

.8867 

.9478 

.8877  .9482 

141 

.8886 

.9487 

.8895 

.9491 

.8904 

.9496 

.8913 

.9500 

.8922 

.9505 

.8931  .9509 

142 

.8940 

.9513 

.8949 

.9518 

.8958 

.9522 

.8967 

.9526 

.8976 

.9531 

.8984  .9535 

143 

.8993 

.9539 

.9002 

.9543 

.9011 

.9548 

.9019 

.9552 

.9028 

.9556 

.9037  .9560 

144 

.9045 

.9564 

.9054 

.9568 

.9062 

.9572 

.9071 

.9576 

.9079 

.9580 

.9087  .9584 

145 

.9096 

.9588 

.9104 

.9592 

.9112 

.9596 

.9121 

.9600 

.9129 

.9604 

.9137  .9608 

146 

.9145 

.9612 

.9153 

.9616 

.9161 

.9620 

.9169 

.9623 

.9177 

.9627 

.9185  .9631 

147 

.9193 

.9635 

.9201 

.9638 

.9209 

.9642 

.9217 

.9646 

.9225 

.9650 

.9233  .9653 

148 

.9240 

.9657 

.9248 

.9660 

.9256 

.9664 

.9263 

.9668 

.9271 

.9671 

.9278  .9675 

149 

.9286 

.9678 

.9293 

.9682 

.9301 

.9685 

.9308 

.9689 

.9316 

.9692 

.9323  .9695 

150 

.9330 

.9699 

.9337 

.9702 

.9345 

.9706 

.9352 

.9709 

.9359 

.9712 

.9366  .9716 

151 

.9373 

.9719 

.9380 

.9722 

.9387 

.9725 

.9394 

.9729 

.9401 

.9732 

.9408  .9735 

152 

.9415 

.9738 

.9422 

.9741 

.9428 

.9744 

.9435 

.9747 

.9442 

.9751 

.9448  .9754 

153 

.9455 

.9757 

.9462 

.9760 

.9468 

.9763 

.9475 

.9766 

.9481 

.9769 

.9488  .9772 

154 

.9494 

.9774 

.9500 

.9777 

.9507 

.9780 

.9513 

.9783 

.9519 

.9786 

.9525  .9789 

155 

.9532 

.9792 

.9538 

.9794 

.9544 

.9797 

.9550 

.9800 

.9556 

.9803 

.9562  .9805 

156 

.9568 

.9808 

.9574 

.9811 

.9579 

.9813 

.9585 

.9816 

.9591 

.9819 

.9597  .9821 

157 

.9603 

.9824 

.9608 

.9826 

.9614 

.9829 

.9619 

.9831 

.9625 

.9834 

.9630  .9836 

158 

.9636 

.9839 

.9641 

.9841 

.9647 

.9844 

.9652 

.9846 

.9657 

.9849 

.9663  .9851 

159 

.9668 

.9853 

.9673 

.9856 

.9678 

.9858 

.9683 

.9860 

.9688 

.9863 

.9693  .9865 

160 

.9698 

.9867 

.9703 

.9869 

.9708 

.9871 

.9713 

.9874 

.9718 

.9876 

.9723  .9878 

161 

.9728 

.9880 

.9732 

.9882 

.9737 

.9884 

.9742 

.9886 

.9746 

.9888 

.9751  .9890 

162 

.9755 

.9892 

.9760 

.9894 

.9764 

.9896 

.9769 

.9898 

.9773 

.9900 

.9777  .9902 

163 

.9782 

.9904 

.9786 

.9906 

.9790 

.9908 

.9794 

.9910 

.9798 

.9911 

.9802  .9913 

164 

.9806 

.9915 

.9810 

.9917 

.9814 

.9919 

.9818 

.9920 

.9822 

.9922 

.9826  .9923 

165 

.9830 

.9925 

.9833 

.9927 

.9837 

.9929 

.9841 

.9930 

.9844 

.9932 

.9848  .9933 

166 

.9851 

.9935 

.9855 

.9937 

.9858 

.9938 

.9862 

.9940 

.9865 

.9941 

.9869  .9943 

167 

.9872 

.9944 

.9875 

.9945 

.9878 

.9947 

.9881 

.9948 

.9885 

.9950 

.9888  .9951 

168 

.9891 

.9952 

.9894 

.9954 

.9897 

.9955 

.9900 

.9956 

.9903 

.9957 

.9905  .9959 

169 

.9908 

.9960 

.9911 

.9961 

.9914 

.9962 

.9916 

.9963 

.9919 

.9965 

.9921  .9966 

170 

.9924 

.9967 

.9927 

.9968 

.9929 

.9969 

.9931 

.9970 

.9934 

.9971 

.9936  .9972 

171 

.9938 

.9973 

.9941 

.9974 

.9943 

.9975 

.9945 

.9976 

.9947 

.9977 

.9949  .9978 

172 

.9951 

.9979 

.9953 

.9980 

.9955 

.9981 

.9957 

.9981 

.9959 

.9982 

.9961  .9983 

173 

.9963 

.9984 

.9964 

.9984 

.9966 

.9985 

.9968 

.9986 

.9969 

.9987 

.9971  .9987 

174 

.9973 

.9988 

.9974 

.9988 

.9976 

.9989 

.9977 

.9990 

.9978 

.9991 

.9980  .9991 

175 

.9981 

.9992 

.9982 

.9992 

.9983 

.9993 

.9985 

.9993 

.9986 

.9994 

.9987  .9994 

176 

.9988 

.9995 

.9989 

.9995 

.9990 

.9996 

.9991 

.9996 

.9992 

.9996 

.9992  .9997 

177 

.9993 

.9997 

.9994 

.9997 

.9995 

.9998 

.9995 

.9998 

.9996 

.9998 

.9996  .9998 

178 

.9997 

.9999 

.9997 

.9999 

.9998 

.9999 

.9998 

.9999 

.9999 

.9999 

.9999  .9999 

179 

.9999 

.9999 

.9999 

.9999 

.9999 

.9999 

.9999 

.9999 

.9999 

.0000 

1.0000  .0000 

126     Table  XI — Factor  Table — Logarithms  of  Primes 

If  ^is  prime,  its  logarithm  is  given.     If  JVis  not  prime,  its  factors  are  given. 


jsr 

1 

S 

7 

9 

10 

0043213738 

0128372247 

0293837777 

0374264979 

11 

3-37 

0530784435 

32-13 

717 

12 

112 

3-41 

1038037210 

3-43 

13 

1172712957 

7-19 

1367205672 

1430148003 

14 

3-47 

11  13 

3-72 

1731862684 

16 

1789769473 

32-17 

1958996524 

3-53 

16 

7-23 

2121876044 

2227164711 

132 

17 

32-19 

2380461031 

3-59 

2528530310 

18 

2576785749 

3-61 

11-17 

33-7 

19 

2810333672 

2855573090 

2944662262 

2988530764 

20 

3-67 

7-29 

32-23 

11-19 

21 

3242824553 

3-71 

7-31 

3-73 

22 

13  17 

3483048630 

3560258572 

3598354823 

23 

3-711 

3673559210 

3-79 

3783979009 

24 

3820170426 

35 

13-19 

3-83 

25 

3996737215 

11-23 

4099331233 

7-37 

26 

32-29 

4199557485 

3-89 

4297522800 

27 

4329692909 

3-7-13 

4424797691 

32-31 

28 

4487063199 

4517864355 

7-41 

172 

29 

3-97 

4668676204 

33-11 

13-23 

30 

7-43 

3-101 

4871383755 

3-103 

31 

4927603890 

4955443375 

5010592622 

11-29 

32 

3  107 

17-19 

3-109 

7-47 

33 

5198279938 

32-37 

5276299009 

3113 

34 

11-31 

73 

5403294748 

5428254270 

35 

33-13 

5477747054 

3-7-17 

5550944486 

36 

192 

3-112 

5646660643 

32-41 

37 

7-53 

•5717088318 

13-29 

5786392100 

38 

3-127 

5831987740 

32-43 

5899496013 

39 

17-23 

3-131 

5987905068 

3-719 

40 

6031443726 

13-31 

11-37 

6117233080 

41 

3-137 

7-59 

3-139 

6222140230 

42 

6242820958 

32-47 

7-61 

311-13 

43 

6:344772702 

6364878964 

19-23 

6424645202 

44 

32-72 

6464037262 

3-149 

6522463410 

45 

11-41 

3-151 

6599162001 

33-17 

46 

6637009254 

6655809910 

6693168806 

7-67 

47 

3-157 

11-43 

32-53 

6803355134 

48 

13-37 

3-7-23 

6875289612 

3-163 

49 

6910814921 

17-29 

7-71 

6981005456 

50 

3  167 

7015679851 

3-132 

7067177823 

51 

7-73 

33-19 

11-47 

3-173 

52 

7168377233 

7185016889 

17-31 

232 

53 

3259 

13-41 

3-179 

72-11 

54 

7331972651 

3-181 

7379873263 

32-6I 

55 

19-29 

7-79 

7458551952 

13-43 

56 

3-11-17 

7505083949 

34-7 

7551122664 

57 

7566361082 

3-191 

7611758132 

3  193 

58 

7-83 

11-53 

7686381012 

19-31 

59 

3-197 

7730546934 

3-199 

7774268224 

60 

7788744720 

32-67 

7831886911 

3-7-29 

61 

13-47 

7874604745 

7902851640 

7916906490 

62 

33-23 

7-89 

311-19 

17-37 

63 

8000293592 

3-211 

72-13 

32-71 

r   64 

8068580295 

8082109729 

8109042807 

11-59 

65 

3-7-31 

8149131813 

32-73 

8188854146 

66 

8202014595 

3-13-17 

23-29 

3-223 

67 

11-61 

8280150642 

8305886687 

7-97 

68 

3-227 

8344207037 

3-229 

13-53 

69 

8394780474 

32-711 

17-41 

3-233 

jsr 

2 
3 
5 
7 
11 

13 
17 
19 
23 
29 
31 
37 
41 
43 
47 
53 
59 
61 
67 
71 
73 
79 
83 
89 
97 

LogN 

301029995664 
477121254720 
698970004336 
845098040014 
041392685158 
113943352307 
230448921378 
278753600953 
361727836018 
462396997899 
491361693834 
568201724067 
612783856720 
633468455580 
672097857936 
724275869601 
770852011642 
785329835011 
826074802701 
851258348719 
863322860120 
897627091290 
919078092376 
949390006645 
986771734266 

1301 
1303 
1307 
1319 
1321 
1327 
1361 
1367 
1373 
1381 
1399 
1409 
1423 
1427 
1429 
1433 
1439 
1447 
1451 
1453 
1459 
1471 
1481 
1483 
1487 
1489 
1493 
1499 
1511 
1523 
1531 
1543 
1549 
1553 
1559 

1142772966 
1149444157 
1162755876 
1202447955 
1209028176 
1228709229 
1338581252 
1357685146 
1376705372 
1401936786 
1458177145 
1489109931 
1532049001 
1544239731 
1550322288 
1562461904 
1580607939 
1604685311 
1616674124 
1622656143 
1640552919 
1676126727 
1705550585 
1711411510 
1723109685 
1728946978 
1740598077 
1758016328 
1792644643 
1826999033 
1849751907 
1883659261 
1900514178 
1911714557 
1928461152 

Factor  Table  — Logarithms  of  Primes 

If  ^ is  a  prime,  its  logarithm  is  given.     If  N  is  not  a  prime,  its  factors  are  given. 


12T 


N 

1 

3 

7 

9 

JV 

roj  js^ 

70 

8457180180 

19-37 

7-101 

8506462352 

1567 

1950689965 

71 

32-79 

23-31 

3-239 

8567288904 

1571 

1961761850 

72 

7  103 

3-241 

8615344109 

36 

1579 

1983821300 

73 

17-43 

8651039746 

11-67 

8686444384 

1583 

1994809149 

74 

31319 

8709888138 

32-83 

7-107 

1597 

2033049161 

75 

8756399370 

3-251 

8790958795 

3-11-23 

1601 

2043913319 

76 

8813846568 

7-109 

13-59 

8859263398 

1607 

2060158768 

77 

3-257 

8881794939 

3  7-37 

19-41 

1609 

2065560441 

78 

11-71 

33-29 

8959747324 

3-263 

1613 

2076343674 

79 

7-113 

13-61 

9014583214 

17-47 

1619 

2092468488 

80 

32-89 

11-73 

3-269 

9079485216 

1621 

2097830148 

81 

9090208542 

3-271 

19-43 

32-7-13 

1627 

2113875529 

82 

9143431571 

9153998352 

9175055096 

9185545306 

1637 

2140486794 

83 

3-277 

72-17 

33-31 

9237619608 

1657 

2193225084 

84 

292 

3-281 

7-112 

3-283 

1663 

2208922492 

85 

23-37 

9309490312 

9329808219 

9339931638 

1667 

2219355998 

86 

3-7-41 

93(^0107957 

3-172 

11-79 

1669 

2224563367 

87 

13-67 

32-97 

94299959ai 

3-293 

1693 

2286569581 

88 

9449759084 

945f)607036 

9479236198 

7-127 

1697 

2296818423 

89 

34-11 

19-47 

3-13-23 

29-31 

1699 

2301933789 

90 

17-53 

3-7-43 

9576072871 

32-101 

1709 

2327420627 

91 

9595183770 

11-83 

7-131 

9633155114 

1721 

2357808703 

92 

3-307 

13-71 

32-103 

9680157140 

1723 

2362852774 

93 

72-19 

3-311 

9717395909 

3-313 

1733 

2387985627 

94 

9735896234 

23-41 

9763499790 

13-73 

1741 

2407987711 

95 

3-317 

9790929006 

311-29 

7-137 

1747 

2422929050 

96 

312 

32-107 

9854264741 

3-1719 

1753 

2437819161 

97 

9872102299 

7  139 

9898945637 

11-89 

1759 

2452658395 

98 

32-109 

9925535178 

3-7-47 

23-43 

1777 

2496874278 

99 

9960736545 

3-331 

9986951583 

33-37 

1783 

2511513432 

100 

7-11-13 

17-59 

19-53 

0038911662 

1787 

2521245525 

101 

3-337 

0056094454 

32-113 

0081741840 

1789 

2526103406 

102 

0090257421 

3-11-31 

13-79 

3-73 

1801 

2555137128 

103 

0132586653 

0141003215 

17-61 

0166155476 

1811 

2579184503 

104 

3-347 

7-149 

3-349 

0207754882 

1823 

2607866687 

105 

0216027160 

34-13 

7-151 

3-353 

1831 

2626883443 

106 

0257153839 

0265332645 

11-97 

0289777052 

1847 

2664668954 

107 

32-7-17 

29-37 

3-359 

13-83 

1861 

2697463731 

108 

23-47 

3-192 

0362295441 

32-112 

1867 

2711443179 

109 

0378247506 

0386201619 

0402066276 

7-157 

1871 

2720737875 

110 

3-367 

0425755124 

33-41 

0449315461 

1873 

2725377774 

111 

11-101 

3-7-53 

0480531731 

3-373 

1877 

2734642726 

112 

19-59 

0503797563 

72-23 

0526939419 

1879 

2739267801 

113 

313-29 

11  103 

3-379 

17-67 

1889 

2762319579 

114 

7  163 

32-127 

31-37 

3-383 

um 

2789821169 

115 

0610753236 

0618293073 

13-89 

19-61 

1^)07 

2803506930 

116 

33-43 

0655797147 

3-389 

7-167 

1913 

2817149700 

117 

0685568951 

3-17-23 

11-107 

32-131 

1931 

2857822738 

118 

0722498976 

7-132 

0744507190 

29-41 

1933 

2862318540 

119 

3-397 

0766404437 

32-7-19 

11-109 

1949 

2898118391 

120 

0795430074 

3-401 

17-71 

3-13-31 

1951 

2902572694 

121 

7-173 

0838608009 

0852905782 

23-53 

1973 

2961270853 

122 

3-11-37 

0874264570 

3-409 

0895518829 

1979 

2964457942 

123 

0902580529 

32-137 

0923696996 

3-7-59 

1987 

2981978671 

124 

17-73 

11-113 

29-43 

0965624384 

1993 

2995072987 

125 

32-139 

7-179 

3-419 

1000257301 

1997 

3003780649 

126 

13-97 

3-421 

7-181 

33-47 

1999 

3008127941 

127 

31-41 

19-67 

1061908973 

1068705445 

2003 

3016809493 

128 

3-7-61 

1082266564 

3211-13 

1102529174 

2011 

3034120706 

129 

1109262423 

3-431 

1129399761 

3-433 

2017 

3047058982 

128  Table  XII  a — Compound  Interest :  ( 1  +  r)»» 

Amount  of  One  Dollar  Principal  at  Compound  Interest  After  n.  Yeabs 


n 

2^10 

2\<^o 

3  fo 

5J% 

4t^o 

4i% 

5^0 

e^o 

7^0 

1 

2 
3 

4 
5 
6 

7 
8 
9 

1.0200 
1.0404 
1.0612 

1.0824 
1.1041 
1.1262 

1.1487 
1.1717 
1.1951 

1.0250 
1.0506 
1.0769 

1.1038 
1.1314 
1.1597 

1.1887 
1.2184 
1.2489 

1.0300 
1.0609 
1.0927 

1.1255 
1.1593 
1.1941 

1.2299 
1.2668 
1.3048 

1.0350 
1.0712 
1.1087 

1.1475 

1.1877 
1.2293 

1.2723 
1.3168 
1.3629 

1.0400 
1.0816 
1.1249 

1.1699 
1.2167 
1.2653 

1.3159 

1.3686 
1.4233 

1.0450 
1.0920 
1.1412 

1.1925 
1.2462 
1.3023 

1.3609 
1.4221 
1.4861 

1.0500 
1.1025 
1.1576 

1.2155 
1.2763 
1.3401 

1.4071 
1.4775 
1.5513 

1.0600 
1.1236 
1.1910 

1.2625 

1.3382 
1.4185 

1.5036 
1.5938 
1.6895 

1.0700 
1.1449 
1.2250 

1.3108 
1.4026 
1.5007 

1.6058 
1.7182 

1 .8385 

10 

1.2190 

1.2801 

1.3439 

1.4106 

1.4802 

1.5530 

1.6289 

1.7908 

1  9672 

11 
12 
13 

14 
15 
16 

17 
18 
19 

1.2434 
1.2682 
1.2936 

1.3195 
1.3459 
1.3728 

1.4002 

1.4282 
1.4568 

1.3121 
1.3449 
1.3785 

1.4130 
1.4483 
1.4845 

1.5216 
1.5597 

1.5987 

1.3842 
1.4258 
1.4685 

1.5126 

1.5580 
1.6047 

1.6528 
1.7024 
1.7535 

1.4600 
1.5111 
1.5640 

1.6187 
1.6753 
1.7340 

1.7947 
1.8575 
1.9225 

1.5395 
1.6010 
1.6651 

1.7317 

1.8009 
1.8730 

1.9479 

2.0258 
2.1068 

1.6229 
1.69.59 
1.7722 

1.8519 
1.9353 
2.0224 

2.1134 

2.2085 
2.3079 

1.7103 
1.7959 
1.8856 

1.9799 

2.0789 
2.1829 

2.2920 

2.4066 
2.5270 

1.8983 
2.0122 
2.1329 

2.2609 
2.3966 
2.5404 

2.6928 
2.8543 
3.0256 

2.1049 
2.2522 
2.4098 

2.5785 
2.7590 
2.9522 

3.1588 
3.3799 
3.6165 

20 

1.4859 

1.6386 

1.8061 

1.9898 

2.1911 

2.4117 

2.6533 

3.2071 

3.8697 

4.1406 
4.4.304 
4.7405 

5.0724 
5.4274 
5.8074 

6.2139 
6.6488 
7.1143 

21 

22 
23 

24 
25 
26 

27 
28 
29 

1.5157 
1.5460 
1.5769 

1.6084 
1.6406 
1.6734 

1.7069 
1.7410 
1.7758 

1.6796 
1.7216 
1.7646 

1.8087 
1.8539 
1.9003 

1.9478 
1.9965 
2.04(;4 

1.8603 
1.91()1 
1.9736 

2.0328 
2.0938 
2.1566 

2.2213 

2.2879 
2.3566 

2.0594 
2.1315 
2.2061 

2.2833 

2.3632 
2.4460 

2.5316 
2.6202 
2.7119 

2.2788 
2.3699 
2.4647 

2.5633 

2.6058 

2.7725 

2.8834 
2.9987 
3.1187 

2.5202 
2.6337 
2.7522 

2.8760 
3.0054 
3.1407 

3.2820 
3.4297 
3.5840 

2.7860 
2.9253 
3.0715 

3.2251 

3.3864 
3.5557 

3.7335 
3.9201 
4.1161 

3.3996 
3.6035 
3.8197 

4.0489 
4.2919 
4.5494 

4.8223 
5.1117 
5.4184 

30 

1.8114 

2.0976 

2.4273 

2.8068 

3.2434 

3.7453 

4.3219 

5.74.35 

7.6123 

31 
32 
33 

34 
35 
36 

37 
38 
39 

1.8476 

1.8845 
1.9222 

1.9607 
1.9999 
2.0399 

2.0807 
2.1223 

2.1647 

2.1500 
2.2038 
2.2589 

2.3153 
2.3732 
2.4325 

2.4933 
2.5557 
2.6196 

2.5001 
2.5751 
2.6523 

2.7319 
2.8139 
2.8983 

2.9852 
3.0748 
3.1670 

2.9050 
3.0067 
3.1119 

3.2209 
3.3336 
3.4503 

3.5710 
3.6960 
3.8254 

3.3731 
3.5081 
3.6484 

3.7943 
3.9461 
4.1039 

4.2681 
4.4388 
4.6164 

3.9139 
4.0900 
4.2740 

4.4664 
4.6673 
4.8774 

5.0969 
5.3262 
5.5659 

5.8164 

4.5380 
4.7649 
5.0032 

5.2533 
5.5160 
5.7918 

6.0814 
6.3855 
6.7048 

6.0881 
6.4534 
6.8406 

7.2510 
7.6861 
8.1473 

8.6361 
9.1543 
9.7035 

10.2857 

8.1451 
8.7153 
9.3253 

9.9781 
10.6766 
11.4239 

12.2236 
13.0793 
13.9948 

40 

2.2080 

2.6851 

3.2620 

3.9593 

4.8010 

7.0400 

14.9745 

41 

42 
43 

44 
45 
46 

47 
48 
49 

50 

2.2522 
2.2972 
2.3432 

2.3901 
2.4379 

2.4866 

2.5363 
2.5871 
2.6388 

2.7522 
2.8210 
2.8915 

2.9638 
3.0379 
3.1139 

3.1917 
3.2715 
3.3533 

3.3599 
3.4607 
3.5645 

3.6715 
3.7816 
3.8950 

4.0119 
4.1323 
4.2562 

4.0978 
4.2413 
4.3897 

4.5433 
4.7024 
4.8669 

5.0373 
5.2136 
5.3961 

4.9931 
5.1928 
5.4005 

5.6165 
5.8412 
6.0748 

6.3178 
6.5705 
6.8333 

6.0781 
6.3516 
6.6374 

6.9361 

7.2482 
7.5744 

7.9153 
8.2715 
8.(i437 

9.0326 

7.3920 
7.7616 
8.1497 

8.5572 
8.9850 
9.4343 

9.9060 
10.4013 
10.9213 

10.9029 
11.5570 
12.2505 

12.9855 
13.7646 
14.5905 

15.4659 
16.3939 
17.3775 

18.4202 

16.0227 
17.1443 
18.3444 

19.6285 
21.0025 
22.4726 

24.0457 
25.7289 
27.5299 

29.4570 

2.6916 

3.4371 

4.3839 

5.5849 

7.1067 

11.4674 

Table  XII  & — Compound  Discount :  1/(1  +  r)"       129 


Present  Value  of 

One  Dollar  Due  at  the  End  of  n  Years 

n 

2^0 

2\^o 

31o 

S\^o 

4=^0 

4:\^0 

S^'/o 

6% 

7% 

1 

2 
3 

4 
5 
6 

7 
8 
9 

10 

.98039 
.96117 
.94232 

.92385 
.90573 
.88797 

.87056 
.85319 
.83676 

.97561 
.95181 
.92860 

.90595 
.88385 
.86230 

.84127 
.82075 
.80073 

.78120 

.97087 
.94260 
.91514 

.88849 
.86261 
.83748 

.81309 
.78941 
.76642 

.96618 
.93351 
.90194 

.87144 
.84197 
.81350 

.78599 
.75941 
.73373 

.96154 
.92456 
.88900 

.85480 
.82193 
.79031 

.75992 
.73069 
.70259 

.95694 
.91573 
.87630 

.83856 
.80245 
.76790 

.73483 
.70319 
.67290 

.95238 
.f)0703 
.86384 

.82270 

.78353 
.74622 

.71068 
.67684 
.64461 

.94340 
.89000 
.83962 

.79209 
.74726 
.70496 

.66506 
.62741 
.59190 

.93458 
.87344 
.81630 

.76290 
.71299 
.66634 

.62275 
.58201 
.54393 

.82035 

.74409 

.70892 

.67556 

.64393 

.61391 

.55839 

.50835 

11 
12 
13 

14 
15 
16 

17 

18 
19 

20 

.80426 
.78849 
.77303 

.75788 
.74301 
.72845 

.71416 
.70016 
.68643 

.76214 
.74356 
.72542 

.70773 
.69047 
.67362 

.65720 
.64117 
.62553 

.72242 
!70138 
.68095 

.66112 
.64186 
.62317 

.60502 
.58739 
.57029 

.68495 
.66178 
.63940 

.61778 
.59689 
.57671 

.55720 
.53836 
.52016 

.64958 
.62460 
.60057 

.57748 
.55526 
.53391 

.51337 
.493()3 
.47464 

.45639 

.61620 
.58966 
.56427 

.53997 
.51672 
.49447 

.47318 
.45280 
.43330 

.58468 
.55684 
.53032 

.50507 

•  .48102 

.45811 

.43630 
.41552 
.39573 

.52679 
.49697 
.46884 

.44230 
.41727 
.39365 

.37136 
.35034 
.33051 

.47509 
.44401 
.41496 

.38782 
.36245 
.33873 

.31657 
.29586 
.27651 

.67297 

.61027 

.55368 

.50257 

.41464 

.37689 

.31180 

.25842 

.24151 
.22571 
.21095 

.19715 
.18425 
.17220 

.16093 
.15040 
.14056 

21 
22 
23 

24 
25 
26 

27 
28 
29 

.65978 
.64684 
.63416 

.62172 
.60953 
.59758 

.58586 
.57437 
.56311 

.59539 
.58086 
.56670 

.55288 
.53939 
.52623 

.51340 

.50088 
.48866 

.53755 
.52189 
.50669 

.49193 
.47761 
.46369 

.45019 
.43708 
.42435 

.48557 
.46915 
.45329 

.43796 
.42315 

.40884 

.39501 
.38165 
.36875 

.43883 
.42196 
.40573 

.39012 
.37512 
.36069 

.34682 
.33348 
.32065 

.39679 
.37970 
.36335 

.34770 
.33273 
.31840 

.30469 
.29157 
.27902 

.35894 
.34185 
.32557 

.31007 
.29530 
.28124 

.26785 
.25509 
.24295 

.29416 
.27751 
.26180 

.24698 
.23300 
.21981 

.20737 
.19563 
.18456 

30 

.55207 

.47674 

.41199 

.35628 

.30832 

.26700 

.23138 

.17411 

.13137 

31 
32 
33 

34 
35 
36 

37 
38 
39 

.54125 
.53063 
.52023 

.51003 
.50003 
.49022 

.48061 
.47119 
.46195 

.46511 
.45377 
.44270 

.43191 
.42137 
.41109 

.40107 
.39128 
.38174 

.39999 
.38834 
.37703 

.36604 
.35538 
.34503 

.33498 
.32523 
.31575 

.34423 
.33259 
.32134 

.31048 
.29998 
.28983 

.28003 
.27056 
.26141 

.29646 
.28506 
.27409 

.26355 
.25:^2 
.24367 

.23430 
.22529 
.21662 

.25550 
.24450 
.23397 

.22390 
.21425 
.20503 

.19620 
.18775 
.17967 

.22036 
.20987 
.19987 

.19035 
.18129 
.17266 

.16444 
.15661 
.14915 

.16425 
.15496 
.14619 

.13791 
.13011 
.12274 

.11580 
.10924 
.10306 

.12277- 
.11474 
.10723 

.10022 
.09366 
.08754 

.08181 
.07646 
.07146 

40 

41 
42 
43 

44 
45 
46 

47 
48 
49 

.45289 

.37243 

.30656 

.25257 

.20829 

.17193 

.14205 

.09722 

.06678 

.44401 
.43530 
.42677 

.41840 
.41020 
.40215 

.39427 
.38654 
.37896 

.36335 
.35448 
.34584 

.33740 
.32917 
.32115 

.31331 
.30567 

.29822 

.29094 

.29763 
.28896 
.28054 

.27237 
.26444 
.25674 

.24926 
.24200 
.23495 

.22811 

.24403 
.23578 
.22781 

.22010 
.21266 
.20547 

.19852 
.19181 
.18532 

.17905 

.20028 
.19257 
.18517 

.17805 
.17120 
.16461 

.15828 
.15219 
.14634 

.16453 
.15744 
.15066 

.14417 
.13796 
.13202 

.12634 
.12090 
.11569 

.11071 

.13528 
.12884 
.12270 

.11686 
.11130 
.10600 

.10095 
.09614 
.09156 

.09172 
.08653 
.08163 

.07701 
.07265 

.06854 

.06466 
.06100 
.05755 

.06241 
.05833 
.05451 

.05095 
.04761 
.04450 

.04159 
.03887 
.03632 

50 

.37153 

.14071 

.08720 

.05429 

.03395 

130 


Table  XII  c—  Amount  of  an  Annuity 


Amount  of 

AN  Annuity  op  One  Dollar 

PER  Year  after  n 

Years 

n 

1 

2 
3 

4 
5 
6 

7 
8 
9 

10 

11 
12 
13 

14 
15 

16 

17 

18 
19 

20 

21 

22 
23 

24 
25 
26 

27 

28 
29 

2'fo 

21^0 

5% 

S\^o 

4% 

4i% 

5^0 

6^0 

7% 

1.0700 
2.2149 
3.4399 

4.7507 
6.1533 
7.6540 

9.2598 
10.9780 
12.8164 

1.0200 
2.0604 
3.1216 

4.2040 
5.3081 
6.4343 

7.5830 
8.7546 
9.9497 

1.0250 
2.0756 
3.1525 

4.2563 

5..3877 
6.5474 

7.7361 

8.9545 

10.2034 

1.0300 
2.0909 
3.1836 

4.3091 
5.4684 
6.6625 

7.8923 

9.1591 

10.4639 

1.0350 
2.1062 
3.2149 

4.3625 
5.5502 
6.7794 

8.0517 

9.3685 

10.7314 

12.1420 

1.0400 
2.1216 
3.2465 

4.4163 
5.6330 
6.8983 

8.2142 

9.5828 

11.0061 

1.0450 
2.1370 
3.2782 

4.4707 
5.7169 
7.0192 

8.3800 
9.8021 
11.2882 

1.0500 
2.1525 
3.3101 

4.5256 
5.8019 
7.1420 

8.5491 
10.0266 
11.5779 

1.0600 
2.1836 
3.3746 

4.6371 
5.9753 
7.3938 

8.8975 
10.4913 
12.1808 

11.1687 

11.4835 

11.8078 

12.4864 

12.8412 

13.2068 

13.9716 

14.7836 

16.8885 
19.1406 
21.5505 

24.1290 

26.8881 
29.8402 

32.9990 
36.3790 
39.9955 

12.4121 
13.6803 
14.9739 

16.2934 
17.6393 
19.0121 

20.4123 
21.8406 
23.2974 

12.7956 
14.1404 
15.5190 

16.9319 
18.3802 
19.8647 

21.3863 
22.9460 
24.5447 

13.1920 
14.6178 
16.0863 

17.5989 
19.1569 
20.7616 

22.4144 
24.1169 
25.8704 

13.6020 
15.1130 
16.6770 

18.2957 
19.9710 
21.7050 

23.4997 
25.3572 

27.2797 

14.0258 
15.6268 
17.2919 

19.0236 
20.8245 
22.6975 

24.6454 
26.6712 

28.7781 

14.4640 
16.1599 
17.9321 

19.7841 
21.7193 
23.7417 

25.8551 
28.0636 
30.3714 

14.9171 
16.7130 
18.5986 

20.5786 
22.6575 
24.8404 

27.1324 
29.5390 
32.0660 

15.8699 
17.8821 
20.0151 

22.2760 
24.6725 
27.2129 

29.9057 
32.7600 
35.7856 

24.7833 

26.1833 

27.6765 

29.2695 

30.9692 

32.7831 

34.7193 

38.9927 

43.8652 

26.2990 
27.8450 
29.4219 

31.0303 
32.6709 
34.3443 

36.0512 
37.7922 
39.5681 

27.8629 
29.5844 
31.3490 

33.1578 
35.0117 
36.9120 

38.8598 
40.8563 
42.9027 

29.5368 
31.4529 
33.4265 

35.4593 
37.5530 
39.7096 

41.9309 
44.2189 
46.5754 

31.3289 
33.4604 
35.6665 

37.9499 
40.3131 
42.7591 

45.2906 
47.9108 
50.6227 

33.2480 
35.6179 
38.0826 

40.6459 
43.3117 
46.0842 

48.9676 
51.9663 
55.0849 

35.3034 
37.9370 
40.6892 

43.5652 
46.5706 
49.7113 

52.9933 
56.4230 
60.0071 

37.5052 
40.4305 
43.5020 

46.7271 
50.1135 
53.6691 

57.4026 
61.3227 
65.4388 

42.3923 
45.9958 
49.8156 

53.8645 
58.1564 
62.7058 

67.5281 
72.6398 

78.0582 

48.0057 
62.4361 
57.1767 

62.2490 
67.6765 
73.4838 

79.6977 
86.3465 
93.4608 

30 

41.3794 

45.0003 

49.0027 

53.4295 

58.3283 

63.7524 

69.7608 

83.8017 

101.0730 

31 
32 
33 

34 
35 
36 

37 
38 
39 

40 

43.2270 
45.1116 
47.0338 

48.9945 
50.9944 
53.0343 

55.1149 
57.2372 
59.4020 

47.1503 
49.3540 
51.6129 

53.9282 
56.3014 
58.7339 

61.2273 
63.7830 
66.4026 

69.0876 

71.8398 
74.6608 
77.5523 

80.5161 
83.5540 
86.6679 

89.8596 
93.1311 
96.4843 

51.5028 
54.0778 
56.7302 

59.4621 
62.2759 
65.1742 

68.1594 
71.2342 
74.4013 

56.3345 
59.3412 
62.4532 

65.6740 
69.0076 
72.4579 

76.0289 
79.7249 
83.5503 

61.7015 
65.2095 
68.8579 

72.6522 

76.5983 
80.7022 

84.9703 
89.4091 
94.0255 

67.6662 
71.7562 
76.0303 

80.4966 
85.1640 
90.0413 

95.1382 
100.4644 
106.0303 

111.8467 

74.2988 
79.0638 
84.0670 

89.3203 

94.8363 

100.6281 

106.7095 
113.0950 
119.7998 

89.8898 

96.3432 

103.1838 

110.4348 
118.1209 
126.2681 

134.9042 
144.0585 
153.7620 

109.2182 
117.9334 

127.2588 

137.2369 
147.9135 
159.3374 

171.5610 
184.6403 
198.6351 

61.6100 

77.6633 

87.5095 

98.8265 

126.8398 

164.0477 

213.6096 

41 
42 
43 

44 
45 
46 

47 
48 
49 

50 

63.8622 
66.1595 
68.5027 

70.8927 
73.3306 
75.8172 

78.3535 
80.9406 
83.5794 

81.0232 
84.4839 
88.0484 

91.7199 
95.5015 
99.3965 

103.4084 
107.5406 
111.7969 

91.6074 

95.8486 

100.2383 

104.7817 
109.4840 
114.3510 

119.3883 
124.6018 
129.9979 

103.8196 
109.0124 
114.4129 

120.0294 
125.8706 
131.9454 

138.2632 
144.8337 
151.6671 

117.9248 
124.2764 
130.9138 

137.8500 
145.0982 
152.6726 

160.5879 
168.8594 
177.5030 

134.2318 
141.9933 
150.1430 

158.7002 
167.6852 
177.1194 

187.0254 
197.4267 
208.3480 

174.9505 
186.5076 
198.7580 

211.7435 
225.5081 
240.0986 

255.5645 
271.9584 
289.3359 

229.6322 
246.7765 
265.1209 

284.7493 
305.7518 
328.2244 

352.2701 
377.9990 
405.5289 

86.2710 

99.9215 

116.1808 

135.5828 

158.7738 

186.5357 

219.8154 

307.7561 

434.9860 

Table  XII  d  —  Present  Value  of  an  Annuity  131 


Present  Value 

OF  One  Dollar  per  Year  for  n  Yeai 

R8 

n 

1 

2 

3 

4 
5 

6 

7 
8 
9 

2^0 

2\^o 

3^0 

Sl^o 

4:<f0 

4:\^o 

S'fo 

6^0 

7fc 

.9804 
l.i>416 

2.8839 

3.8077 
4.7135 
5.6014 

6.4720 
7.3255 
8.1622 

.9756 
1.9274 
2.8560 

3.7620 
4.6458 
5.5081 

6.3494 
7.1701 
7.9709 

.9709 
1.9135 

2.8286 

3.7171 
4.5797 
5.4172 

6.2303 
7.0197 

7.7861 

.9662 
1.8997 
2.8016 

3.6731 
4.5151 
5.3286 

6.1145 
6.8740 
7.6077 

8.3166 

.9615 
1.88(U 
2.7751 

3.6299 
4.4518 
5.2421 

6.0021 
6.7327 
7.4353 

.9569 
1.87-:7 
2.7490 

3.5875 
4.3900 
5.1579 

5.8927 
6.5959 

7.2688 

.9524 
1.8594 
2.7232 

3.5460 
4.3295 
5.0757 

5.7864 
6.4632 
7.1078 

7.7217 

.9434 
1.8334 
2.6730 

3.4651 
4.2124 
4.9173 

5.5824 
6.2098 
6.8017 

.9346 
1.8080 
2.6243 

3.3872 
4.1002 
4.7665 

5.3893 
5.9713 
6.5152 

10 

8.9826 

8.7521 

8.5302 

8.1109 

7.9127 

7.3601 

7.0236 

11 
12 
13 

14 
15 
16 

17 
18 
19 

9.7868 
10.5753 
11.3484 

12.1062 
12.8493 
13.5777 

14.2919 
14.9920 
15.6785 

9.5142 
10.2578 
10.9832 

11.6909 
12.3814 
13.0550 

13.7122 
14.3534 
14.9789 

9.2526 

9.9540 

10.6350 

11.2961 
11.9379 
12.5611 

13.1661 
13.7535 
14.3238 

9.0016 

9.6633 

10.3027 

10.9205 
11.5174 
12.0941 

12.6513 

13.1897 
13.7098 

8.7605 
9.3851 
9.9856 

10.5631 
11.1184 
11.6523 

12.1657 
12.6593 
13.1339 

8.5289 
9.1186 
9.6829 

10.2228 
10.7395 
11.2340 

11.7072 
12.1600 
12.5933 

8.3064 
8.8633 
9.3936 

9.8986 
10.3797 
10.8378 

11.2741 

11.6896 
12.0853 

7.8869 
8.3838 
8.8527 

9.2950 

9.7122 

10.1059 

10.4773 
10.8276 
11.1581 

7.4987 
7.9427 
8.3577 

8.7455 
9.1079 
9.4466 

9.7632 
10.0591 
10.3356 

20 

21 
22 
23 

24 
25 
26 

27 
28 
29 

30 

16.3514 

15.5892 

14.8775 

14.2124 

13.5903 

14.0292 
14.4511 

14.8568 

15.2470 
15.6221 
15.9828 

16.3296 
16.6631 
16.9837 

13.0079 

12.4622 

11.4699 

10.5940 

17.0112 
17.6580 
18.2922 

18.9139 
19.5235 
20.1210 

20.7069 
21.2813 
21.8444 

16.1845 
16.7654 
17.3321 

17.8850 
18.4244 
18.9506 

19.4640 
19.9649 
20.4535 

15.4150 
15.9369 
16.4436 

16.9355 
17.4131 

17.8768 

18.3270 
18.7641 
19.1885 

14.6980 
15.1671 
15.6204 

16.0584 
16.4815 
16.8904 

17.2854 
17.6670 
18.0358 

13.4047 
13.7844 
14.1478 

14.4955 

14.8282 
15.1466 

15.4513 
15.7429 
16.0219 

16.2889 

16.5444 

16.7889 
17.0229 

17.2468 
17.4610 
17.6660 

17.8622 
18.0500 
18.2297 

12.8212 
13.1630 
13.4886 

13.7986 
14.0939 
14.3752 

14.6430 
14.8981 
15.1411 

11.7641 
12.0416 
12.3034 

12.5504 
12.7834 
13.0032 

13.2105 
13.4062 
13.5907 

10.8355 
11.0612 
11.2722 

11.4693 
11.6536 

11.8258 

11.9867 
12.1371 

12.2777 

22.3965 

20.9303 

19.6004 

18.3920 

18.7363 
19.0689 
19.3902 

19.7007 
20.0007 
20.2905 

20.5705 
20.8411 
21.1025 

21.3551 

21.5991 
21.8349 
22.0627 

22.2828 
22.4955 
22.7009 

22.8994 
23.0912 
23.2766 

17.2920 

15.3725 

13,7648 

12.4090 

31 
32 
33 

34 
35 

36 

37 
38 
39 

40 

22.9377 
23.4683 
23.9886 

24.4986 
24.9986 
25.4888 

25.9695 
26.4406 
26.9026 

21.3954 
21.8492 
22.2919 

22.7238 
23.1452 
23.5563 

23.9573 
24.3486 
24.7303 

20.0004 
20.3888 
20.7658 

21.1318 
21.4872 
21.8323 

22.1672 
22.4925 

22.8082 

17.5885 
17.8736 
18.1476 

18.4112 
18.6646 
18.9083 

19.1426 
19.3679 
19.5845 

15.5928 
15.8027 
16.0025 

16.1929 
16.3742 
16.5469 

16.7113 
16.8679 
17.0170 

13.9291 
14.0840 
14.2302 

14.3681 
14.4982 
14.6210 

14.7368 
14.8460 
14.9491 

12.5318 
12.6466 
12.7538 

12.8540 
12.9477 
13.0352 

13.1170 
13.1935 
13.2649 

27.3555 

25.1028 

23.1148 

19.7928 

18.4016 

17.1591 

15.0463 

15.1380 
15.2245 
15.3062 

15.3832 
15.4558 
15.5244 

15.5890 
15.6500 
15.7076 

13.3317 

41 
42 
43 

44 
45 
46 

47 
48 
49 

50 

27.7995 
28.2348 
28.6616 

29.0800 
29.4902 
29.8923 

30.2866 
30.6731 
31.0521 

25.4661 
25.8206 
26.1664 

26.5038 
26.8330 
27.1542 

27.4675 
27.7732 
28.0714 

23.4124 
23.7014 
23.9819 

24.2543 
24.5187 
24.7754 

25.0247 
25.2667 
25.5017 

19.9931 
20.1856 
20.3708 

20.5488 
20.7200 
20.8847 

21.0429 
21.1951 
21.3415 

18.5661 
18.7236 

18.8742 

19.0184 
19.1563 
19.2884 

19.4147 
19.5356 
19.6513 

17.2944 
17.4232 
17.5459 

17.6628 
17.7741 
17.8801 

17.9810 
18.0772 
18.1687 

13.3941 
13.4524 
13.5070 

13.5579 
13.6055 
13.6500 

13.6910 
13.7305 
13.7668 

31.4236 

28.3623 

25.7298 

23.4556 

21.4822 

19.7620 

18.2559 

15.7619 

13.8007 

132  Table  XII  e  —  Logarithms  for  Interest  Computations 


r 

1-^r 

log  {1  +  r) 

\% 

1.005 

00216  60617  56508 

1% 

1.010 

00432  13737  82643 

U% 

1.015 

00646  60422  49232 

2% 

1.020 

00860  01717  61918 

2^% 

1.025 

01072  38653  91773 

3% 
3^% 

1.030 

01283  72247  05172 

1.035 

01494  03497  92937 

4% 

1.040 

01703  33392  98780 

4^% 

1.045 

01911  62904  47073 

6% 

1.050 

02118  92990  69938 

r 

1  +r 

log  (1  +  r) 

5h% 

1.055 

02325  24596  33711 

6% 

1.060 

02530  58652  64770 

6^% 

1.065 

02734  96077  74757 

7% 

1.070 

02938  37776  85210 

7i% 

1.075 

03140  84642  51624 

8% 

1.080 

03342  37554  86950 

Sh% 

1.085 

03542  97381  84548 

9% 

1.090 

03742  64979  40624 

91% 

1.095 

03941  41191  76137 

10% 

1.100 

04139  26851  58225 

For  Amount,  A,  of  any  principal,  P,  after  n  years :  A  =  P  (l-\-  r)n 
For  present  worth,  P,  of  any  amount.  A,  at  the  end  of  n  years:  P  =  A-i-  (l-\-r)n 
To  find  logarithms  and  antilogarithms  of  A  and  P  to  many  significant  figures,  use 
Table  XI,  p.  126,  and  Table  I  a,  p.  20. 


TABLE  XII /—AMERICAN  EXPERIENCE  MORTALITY  TABLE 

Based  on  100,000  living  at  age  10 


At 
Age 

10 

Number 
Surviving 

Deaths 

At 
Age 

Number 
Surviving 

Deaths 

At 
Age 

Number 
Surviving 

Deaths 

At 
Age 

Number 
Surviving 

Deaths 

100,000 

749 

35 

81,822 

732 

60 

57,917 

1,546 

85 

5,485 

1,292 

11 

99,251 

746 

36 

81,090 

737 

61 

56,371 

1,628 

86 

4,193 

1,114 

12 

98,505 

743 

37 

80,353. 

742 

62 

54,743 

1,713 

87 

3,079 

933 

13 

97,762 

740 

38 

79,611 

749 

63 

53,030 

1,800 

88 

2,146 

744 

14 

97,022 

737 

39 

78,862 

756 

64 

51,230 

1,889 

89 

1,402 

555 

15 

96,285 

735 

40 

78,106 

765 

65 

49,341 

1,980 

90 

847 

385 

16 

95,550 

732 

41 

77,341 

774 

66 

47,361 

2,070 

91 

462 

246 

17 

94,818 

729 

42 

76,567 

785 

67 

45,291 

2,158 

92 

216 

137 

18 

94,089 

727 

43 

75,782 

797 

68 

43,133 

2,243 

93 

79 

58 

19 

93,362 

725 

44 

74,985 

812 

69 

40,890 

2,321 

94 

21 

18 

20 

92,637 

723 

45 

74,173 

828 

70 

38,569 

2,391 

95 

3 

3 

21 

91,914 

722 

46 

73,345 

848 

71 

36,178 

2,448 

22 

91,192 

721 

47 

72,497 

870 

72 

33,730 

2.487 

23 

90,471 

720 

48 

71,627 

896 

73 

31,243 

2,505 

24 

89,751 

719 

49 

70,731 

927 

74 

28,738 

2,501 

25 

89,032 

718 

50 

69,804 

962 

75 

26,237 

2,476 

26 

88,314 

718 

51 

68,842 

1,001 

76 

23,761 

2,431 

27 

87,596 

718 

52 

67,841 

1,044 

77 

21,330 

2,369 

28 

86,878 

718 

53 

66,797 

1,091 

78 

18,961 

2,291 

29 

86,160 

719 

54 

65,706 

1,143 

79 

16,670 

2,196 

30 

85,441 

720 

55 

64,563 

1,199 

80 

14,474 

2,091 

31 

84,721 

721 

56 

63,364 

1,260 

81 

12,383 

1,964 

32 

84,000 

723 

57 

62,104 

1,325 

82 

10,419 

1,816 

33 

83,277 

726 

58 

60,779 

1,394 

83 

8,603 

1,648 

34 

82,551 

729 

59 

59,385 

1,468 

84 

6,955 

1,470 

XIII] 


Table  XIII  —  Important  Constants 

Logarithms  of  Important  Constants 


133 


n  —  NUMBER 

Value  of  n 

LoGio  n 

IT 

3.14159265 

0.49714987 

l-^TT 

0.31830989 

9.50285013 

7r2 

9.86960440 

0.99429975 

v^ 

1.77245385 

0.24857494 

e  =  Naperian  Base 

2.71828183 

0.43429448 

M  =  logio  e 

0.43429448 

9.63778431 

l^.¥=logelO 

2.30258509 

0.36221569 

180  -f-  TT  =  degrees  in  1  radian 

57.2957795 

1  75812263 

TT  ^  180  =  radians  in  1° 

0.01745329 

8.24187737 

w  -f- 10800  =  radians  in  1' 

0.0002908882 

6.46372612 

TT  -^  648000  =  radians  in  1" 

0.000004848136811095 

4.68557487 

sin  1" 

0.000004848136811076 

4.68557487 

tan  1" 

0.000004848136811152 

4.68557487 

centimeters  in  1  ft. 

30.480 

1.4840158 

feet  in  1  cm. 

0.032808 

8.5159842 

inches  in  1  m. 

39.37  (exact  legal  value) 

1.5951654 

pounds  in  1  kg. 

2.20462 

0.3433340 

kilograms  in  1  lb. 

0.453593 

9.6566660 

g  (average  value) 

32.16  ft./sec./sec. 
=  981  cm. /sec. /sec. 

1.5073 
2.9916690 

weight  of  1  cu.  ft.  of  water 

62.425  lb.  (max.  density) 

1.7953586 

weight  of  1  cu.  ft.  of  air 

0.0807  lb.  (at  32°  F.) 

8.907 

cu.  in.  in  1  (U.  S.)  gallon 

231  (exact  legal  value) 

2.3636120 

ft.  lb.  per  sec.  in  1  H.  P. 

550  (exact  legal  value) 

2.7403627 

kg.  m.  per  sec.  in  1  H.  P. 

76.0404 

1.8810445 

watts  in  1  H.  P. 

745.957 

2,8727135 

Several  Numbers  Very  Accurately 


TT  =  3.14159 

26535 

89793 

23846 

26433 

83280 

e  =  2.71828 

18284 

59045 

23536 

02874 

71353 

3/ =0.43429 

44819 

03251 

82765 

11289 

18917 

1 --3/ =2.30258 

50929 

94045 

68401 

79914 

54684 

logio  TT  =  0.49714 

98726 

94133 

85435 

12682 

88291 

logio  M  =  9.63778 

43113 

00536 

78912 

Certain  Convenient  Values  for 


1  TO  n  =  10 


n 

\/n 

"nAJ' 

%fn 

n\ 

\/n\ 

LoGio  n 

1 

1.000000 

1.00000 

1.00000 

1 

1.0000000 

0.000000000 

2 

0.500000 

1.41421 

1.25992 

2 

0.5000000 

0.301029996 

3 

0.333333 

1.73205 

1.44225 

6 

0.1666667 

0.477121255 

4 

0.250000 

2.00000 

1.58740 

24 

0.0416667 

0.602059991 

5 

0.200000 

2.23607 

1.70998 

120 

0.0083333 

0.698970004 

6 

0.166667 

2.44949 

1.81712 

720 

0.0013889 

0.778151250 

7 

9.142857 

2.64575 

1.91293 

5040 

0.0001984 

0.845098040 

8 

3.125000 

2.82843 

2.00000 

40320 

0.0000248 

0.903089987 

9 

0.111111 

3.00000 

2.08008 

362880 

0.0000028 

0.954242509 

10 

0.100000 

3.16228 

2.15443 

3628800 

0.0000003 

1.000000000 

134 

Table  XIV 

a- 

-  Four  Place  Logarithms 

[XIV 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

12  3   4  6  6 

1 
7  8  9 

10 

0000 

0043 

0086 

0128 

0170 

0212 

0253 

0294 

0334 

0374 

4  8  12  17  2125 

29  33  37 

11 
12 
13 

14 
15 

16 

17 
18 
19 

0414 
0792 
1139 

1461 
1761 
2041 

2304 
2553 

2788 

0453 
0828 
1173 

1492 
1790 
2068 

2330 

2577 
2810 

0492 

0864 
1206 

1523 
1818 
2095 

2355 
2601 
2833 

0531 
0899 
1239 

1553 
1847 
2122 

2380 
2625 
2856 

0569 
0934 
1271 

1584 

1875 
2148 

2405 
2648 

2878 

0607 
0969 
1303 

1614 
1903 
2175 

2430 
2672 
2900 

0645 
1004 
1335 

1644 
1931 
2201 

2465 
2695 
2923 

0682 
1038 
1367 

1673 
1959 
2227 

2480 

2718 
2945 

0719 
1072 
1399 

1703 

1987 
2253 

2504 

2742 
2967 

0755 
1106 
1430 

1732 
2014 
2279 

2529 
2765 
2989 

4  8  11 
3  7  10 
3  6  10 

3  6  9 
3  6  8 
3  5  8 

2  6  7 
2  5  7 

2  4  7 

15  19  23 
14  17  21 
13  16  19 

12  15  18 
11  14  17 
11  13  16 

10  12  15 
9  12  14 
9  11  13 

26  30  34 
24  28  31 
23  26  29 

21  24  27 
20  22  25 
18  21  24 

17  20  22 
16  19  21 
\i^  18  20 

20 

3010 

3032 

3054 

3075 

3096 

3118 

3139 

3160 

3181 

3201 

2  4  6 

8  1113 

15  17  19 

21 
22 
23 

24 
25 

26 

27 

28 
29 

3222 
3424 
3617 

3802 
3979 
4150 

4314 
4472 
4624 

3243 
3444 
3636 

3820 
3997 
4166 

4330 

4487 
4639 

3263 
3464 
3655 

3838 
4014 
4183 

4346 

4502 
4654 

3284 
3483 
3674 

3856 
4031 
4200 

4362 
4518 
4669 

3304 
3502 
3692 

3874 
4048 
4216 

4378 
4533 
4683 

3324 
3522 
3711 

3892 
4065 
4232 

4393 
4548 
4698 

3345 
3541 
3729 

3909 

4082 
4249 

4409 
4564 
4713 

3365 
3560 
3747 

3927 
4099 
4265 

4425 
4579 

4728 

3385 
3579 
3766 

3946 
4116 
4281 

4440 
4594 
4742 

3404 
3598 
3784 

3962 
4133 

4298 

4456 
4609 

4757 

2  4  6 
2  4  6 
2  4  6 

2  4  6 
2  4  5 
2  3  5 

2  3  6 
2  3  5 
13  4 

8  10  12 
8  10  12 
7  9  11 

7  9  11 
7  9  10 
7  8  10 

6  8  9 
6  8  9 
6  7  9 

14  16  18 
14  16  17 
13  15  17 

12  14  16 
12  14  16 
11  13  15 

11  12  14 
11  12  14 
10  12  13 

30 

4771 

4786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

41^)00 

13  4 

6  7  9 

10  11 13 

31 
32 
33 

34 
35 

36 

37 
38 
39 

4914 
5051 
5185 

5315 
5441 
5563 

5682 
5798 
5911 

4928 
5065 
5198 

5328 
5453 
5575 

5694 
5809 
6922 

4942 
5079 
5211 

5340 
5465 
5587 

6705 
5821 
5933 

4955 
5092 
5224 

5353 

5478 
5599 

5717 
6832 
5944 

4969 
5105 
6237 

6366 
6490 
5611 

5729 
5843 
5955 

4983 
5119 
6260 

6378 
6502 
6623 

6740 
5855 
5966 

4997 
5132 
6263 

5391 
5514 
5635 

6762 
5866 
5977 

5011 
5145 
5276 

6403 

5527 
6647 

6763 

5877 
6988 

6024 
5159 
6289 

6416 
5539 
6658 

5776 

6888 
6999 

5038 
5172 
5302 

5428 
5551 
6670 

6786 
5899 
6010 

13  4 
13  4 
13  4 

12  4 
12  4 
12  4 

12  4 
1  2  3 
1  2  3 

5  7  8 
5  7  8 

5  7  8 

6  6  8 
5  6  7 

5  6  7 

6  6  7 
6  6  7 
4  5  7 

10  11  12 
91112 
9  1112 

9  10  11 
9  10  11 
8  10  11 

8  9  11 
8  9  10 
8  9  10 

40 

6021 

6031 

6042 

6053 

6064 

6076 

6085 

6096 

6107 

6117 

12  3 

4  5  6 

8  9  10 

41 
42 
43 

44 
45 

46 

47 

48 
49 

6128 
6232 
6335 

6435 
6532 
6628 

6721 
6812 
6902 

6138 
6243 
6345 

6444 
6542 
6637 

6730 

6821 
6911 

6149 
6253 
6355 

6454 
6551 
6646 

6739 
6830 
6920 

6160 
6263 
6365 

6464 
6561 
6656 

6749 
6839 
6928 

6170 
6274 
6375 

6474 
6571 
6665 

6758 
6848 
6937 

6180 
6284 
6385 

6484 
6580 
6675 

6767 
6857 
6946 

6191 
6294 
6395 

6493 
6590 
6684 

6776 
6866 
6955 

6201 
6304 
6405 

6503 
6599 
6693 

6785 
6875 
6964 

6212 
6314 
6415 

6513 
6609 
6702 

6794 

6884 
6972 

62?? 
6325 
6425 

6522 
6618 
6712 

6803 
6893 
6981 

12  3 
12  3 
12  3 

12  3 
12  3 
12  3 

12  3 
12  3 
12  3 

4  5  6 
4  6  6 
4  5  6 

4  6  6 
4  5  6 
4  5  6 

4  5  6 
4  6  6 
4  4  6 

7  8  9 
7  8  9 
7  8  9 

7  8  9 
7  8  9 

7  7  8 

7  7  8 
7  7  8 
6  7  8 

50 

6990 

6998 

7007 

7016 

7024 

7033 

7042 

7050 

7059 

7067 

12  3 

3  4  5 

6  7  8 

51 
52 
63 

54 

7076 
7160 
7243 

7324 

7084 
7168 
7251 

7332 

7093 

7177 
7259 

7340 

7101 
7185 
7267 

7348 

7110 
7193 

7275 

7356 

7118 
7202 
7284 

7364 

7126 
7210 
7292 

7372 

7135 

7218 
7300 

7380 

7143 
7226 
7308 

7388 

7152 
7235 
7316 

7396 

1  2  3 
12  3 
12  2 

12  2 

3  4  5 
3  4  6 
3  4  5 

3  4  6 

6  7  8 
6  7  7 
6  6  7 

6  6  7 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

1  2  2 

4  5  6 

7  8  9 

The  proportional  parts  are  stated  in  full  lor  every  tenth  at  the  right-hand  side. 
Xh%  logarithm  of  any  number  of  four  significant  figures  can  be  read  directly  by  add- 


XIVJ 

Table  XIV 

a- 

-  Four  Place  Logarithms 

135 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

» 

12  3 

4  5  6 

7  8  9 

55 

56 

57 
58 
59 

60 

()1 
62 
(53 

(;4 
65 

6(i 

67 
69 

7404 

7482 

7559 
7634 

7709 

7782 

7412 
7490 

7566 
7642 
7716 

7789 

7419 
7497 

7574 
7649 
7723 

7796 

7427 
7505 

7582 
7657 
7731 

7435 
7513 

7589 
7664 

7738 

7443 
7520 

7597 
7672 

7745 

7451 

7528 

7604 
7679 
7752 

7459 
7536 

7612 
7686 
7760 

7466 
7543 

7619 
7694 
7767 

7474 
7551 

7627 
7701 

7774 

1  2  2 
12  2 

112 
1  1  2 
112 

3  4  5 
3,  4  5 

3  4  5 
3  4  4 
3  4  4 

5  6  7 
5  6  7 

5  6  7 
5  6  7 

5  6  7 

7803 

7810 

7818 

7825 

7832 

7839 

7846 

1  1  2 

3  4  4 

5  6  6 

7853 
7924 
7993 

8062 
8129 
8195 

8261 
8325 

8388 

7860 
7931 
8000 

8069 
8136 
8202 

8267 
8331 
8395 

7868 
7938 
8007 

8075 
8142 
8209 

8274 
8338 
8401 

7875 
7945 
8014 

8082 
8149 
8215 

8280 
8344 

8407 

7882 
7952 
8021 

8089 
8156 
8222 

8287 
8351 
8414 

7889 
7959 
8028 

8096 
8162 
8228 

S29S 
8357 
8420 

7896 
7966 
8035 

8102 
8169 
8235 

8299 
8363 
8426 

7903 
7973 
8041 

8109 
8176 
8241 

8306 
8370 
8432 

7910 
7980 
8048 

8116 

8182 
8248 

8312 
8376 
8439 

7917 
7987 
8055 

8122 
8189 
8254 

8319 
8382 
8445 

112 
112 
112 

112 
112 
112 

112 
112 
112 

3  3  4 
3  3  4 
3  3  4 

3  3  4 
3  3  4 
3  3  4 

3  3  4 
3  3  4 
3  3  4 

5  6  6 

5  5  6 

6  6  6 

5  5  6 
5  5  6 
5  5  6 

5  5  6 
4  5  6 

4  5  6 

70 

71 

72 
73 

74 
75 

76 

77 
78 

79 

8451 

8457 

8463 

8470 

8476 

8482 

8488 

8494 

8500 

8506 

112 

3  3  4 

4  5  6 

8513 
8573 
8633 

8692 
8751 
8808 

8865 
8921 
8976 

8519 
8579 
8639 

8698 
8756 
8814 

8871 
8927 
8982 

8525 
8585 
8645 

8704 
8762 
8820 

8876 
8932 
8987 

8531 
8591 
8651 

8710 
8768 
8825 

8882 
8938 
8993 

8537 
8597 
8657 

8716 
8774 
8831 

8887 
8943 
8998 

8543 
8603 
8663 

8722 
8779 
8837 

8893 
8949 
9004 

8549 
8(309 
8669 

8727 
8785 
8842 

8899 
8954 
9009 

8555 
8615 
8675 

8733 
8791 
8848 

8904 
89()0 
9015 

8561 
8621 
8681 

8739 
8797 
8854 

8910 
8965 
9020 

8567 
8627 
8686 

8745 
8802 
8859 

8915 
8971 
9025 

112 
112 
112 

112 
112 
112 

1  1  2 
112 
112 

3  3  4 
3  3  4 
2  3  4 

2  3  4 
2  3  3 
2  3  3 

2  3  3 
2  3  3 
2  3  3 

4  5  6 
4  5  6 
4  5  5 

4  5  5 
4  5  6 
4  4  5 

4  4  5 
4  4  5 
4  4  5 

80 

9031 

9036 

9042 

9047 

9053 

9058 

9063 

9069 

9074 

9079 

1  1  2 

2  3  3 

4  4  5 

81 
82 
83 

84 
85 

86 

87 
88 
89 

9085 
9138 
9191 

9243 
9294 
9345 

9395 
9445 
9494 

9090 
9143 
9196 

9248 
9299 
9350 

9400 
9450 
9499 

9096 
9149 
9201 

9253 
9304 
9355 

9405 
9455 
9504 

9101 
9154 
9206 

9258 
9309 
9360 

9410 
9460 
9509 

9106 
9159 
9212 

9263 
9315 
9365 

9415 
9465 
9513 

9112 
9165 
9217 

9269 
9320 
9370 

9420 
9469 
9518 

9117 
9170 
9222 

9274 
9325 
9375 

9425 
9474 
9523 

9122 
9175 
9227 

9279 
9330 
9380 

9430 
9479 
9528 

9128 
9180 
9232 

9284 
9335 
9385 

9435 
9484 
9533 

9133 
9186 
9238 

9289 
9340 
9390 

9440 
9489 
9538 

112 
112 
112 

112 
112 
112 

112 
Oil 
0  1  1 

2  3  3 
2  3  3 
2  3  3 

2  3  3 
2  3  3 
2  3  3 

2  3  3 
2  2  3 

2  2  3 

4  4  5 
4  4  5 
4  4  5 

4  4  5 
4  4  5 
4  4  5 

4  4  5 
3  4  4 
3  4  4 

90 

9542 

9547 

9552 

9557 

9562 

9566 

9571 

9576 

9581 

9586 

Oil 

2  2  3 

3  4  4 

91 
92 
93 

94 
95 

97 
98 
99 

9590 
9638 
9685 

9731 

9777 
9823 

9868 
9912 
9956 

9595 
9643 

9689 

9736 

9782 
9827 

9872 
9917 
9961 

9600 
9647 
9694 

9741 

9786 
9832 

9877 
9921 
9965 

9605 
9652 
9699 

9745 
9791 
9836 

9881 
9926 
9969 

9609 
9657 
9703 

9750 
9795 
9841 

9886 
9930 
9974 

9614 
9661 
9708 

9754 
9800 
9845 

9890 
9934 
9978 

9619 
9666 
9713 

9759 
9805 
9850 

9894 
9939 
9983 

9624 
9671 
9717 

9763 
9809 
9854 

9899 
9943 

91^87 

9628 
9675 
9722 

9768 
9814 
9859 

9903 
9948 
9991 

9633 
9680 
9727 

9773 

9818 
9863 

9908 
9952 
9996 

0  1  1 
Oil 
Oil 

Oil 
Oil 
Oil 

Oil 
0  1  1 
Oil 

2  2  3 
2  2  3 
2  2  3 

2  2  3 
2  2  3 
2  2  3 

2  2  3 
2  2  3 
2  2  3 

3  4  4* 
3  4  4 
3  4  4 

3  4  4 
3  4  4 
3  4  4 

3  1  4 
3  3  4 
3  3  4 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

1  2  3 

4  5  6 

7  8  9 

mg  the  proportional  part  corresponding  to  the  fourth  figure  to  the  tahular  numhei 
corresponding  to  the  first  three  figures.    There  may  be  an  error  of  1  in  the  last  place. 


136 

Table  XIV  &- 

■  Antilogarithms  to  Four  Places 

[xiy 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

12  3 

4  5  6 

7  8  9 

.00 

1000 

1002 

1005 

1007 

1009 

1012 

1014 

1016 

1019 

1021 

0  0  1 

111 

2  2  2 

.01 
.02 
.03 

.04 
.05 

.06 

.07 
.08 
.09 

.10 

.11 
.12 
.13 

.14 
.15 

.16 

.17 
.18 
.19 

1023 
1047 
1072 

1096 
1122 
1148 

1175 
1202 
1230 

1026 
1050 
1074 

1099 
1125 
1151 

1178 
1205 
1233 

1028 
1052 
1076 

1102 
1127 
1153 

1180 
1208 
1236 

10:30 
1054 
1079 

1104 
1130 
1156 

1183 
1211 
1239 

1033 
1057 
1081 

1107 
1132 
1159 

1186 
1213 
1242 

1035 
1059 
1084 

1109 
1135 
1161 

1189 
1216 
1245 

1038 
1062 
1086 

1112 
1138 
1164 

1191 
1219 
1247 

1040 
1064 
1089 

1114 
1140 
1167 

1194 
1222 
1250 

1042 
1067 
1091 

1117 
1143 
1169 

1197 
1225 
1253 

1045 
1069 
1094 

1119 
1146 
1172 

1199 
1227 
125(> 

0  0  1 
0  0  1 
0  0  1 

Oil 

oil 
oil 

oil 
oil 
oil 

111 
111 
1  1  1 

112 
112 
112 

112 
112 
112 

2  2  2 
2  2  2 

2  2  2 

2  2  2 
2  2  2 

2  2  2 

2  2  2 

2  2  3 
2  2  3 

1259 

1288 
1318 
1349 

1380 
1413 
1445 

1479 
1514 
1549 

1262 

1291 
1321 
1352 

1384 
1416 
1449 

1483 
1517 
1552 

1265 

1294 
1324 
1355 

1387 
1419 
1452 

1486 
1521 
1556 

1268 

1297 
1327 
1358 

1390 
1422 
1455 

1489 
1524 
1560 

1271 

1274 

1276 

1279 

1282 

1285 

0  1  1 

1  1  2 

2  2  3 

1300 
1330 
1361 

1393 
1426 
1459 

1493 
1528 
1563 

1303 
1334 
1365 

1396 
1429 
1462 

1496 
1531 
1567 

1306 
1337 
1368 

1400 
1432 
1466 

1500 
1535 
1570 

1309 
1340 
1371 

1403 
1435 
1469 

1503 

1538 
1574 

1312 
1343 
1374 

1406 
1439 
1472 

1507 
1542 
1578 

1315 
1346 
1377 

1409 
1442 
1476 

1510 
1545 

1581 

oil 

0  1  1 

oil 
oil 

0  1  1 

oil 

oil 
oil 

0  1  1 

12  2 
12  2 
12  2 

12  2 
12  2 
12  2 

1  2  2 
12  2 
12  2 

2  2  3 
2  2  3 
2  3  3 

2  3  3 
2  3  3 
2  3  3 

2  3  3 
2  3  3 
2  3  3 

.20 

.21 
.22 
.23 

.24 
.25 

.26 

.27 

.28 
.29 

1585 

1589 

1626 
1663 
1702 

1742 
1782 
1824 

1866 
1910 
1954 

1592 

1596 

1600 

1603 

1607 

1611 

1614 

1618 

oil 

1  2  2 

3  3  3 

1622 
1660 
1698 

1738 
1778 
1820 

1862 
1905 
1950 

1629 
1667 
1706 

1746 
1786 
1828 

1871 
1914 
1959 

1633 
1671 
1710 

1750 
1791 
1832 

1875 
1919 
1963 

1637 
1675 
1714 

1754 
1795 
1837 

1879 
1923 
1968 

1641 
1679 
1718 

1758 
1799 
1841 

1884 
1928 
1972 

1644 
1683 
1722 

1762 
1803 
1845 

1888 
1932 
1977 

1648 
1687 
1726 

1766 
1807 
1849 

1892 
1936 
1982 

1652 
1690 
1730 

1770 
1811 
1854 

1897 
1941 
1986 

1656 
1694 
1734 

1774 

1816 
1858 

1901 
1945 
1991 

0  1  1 

oil 
oil 

oil 
oil 
oil 

oil 
oil 
oil 

12  2 
2  2  2 
2  2  2 

2  2  2 
2  2  3 
2  2  3 

2  2  3 
2  2  3 
2  2  3 

3  3  3 
3  3  3 
3  3  3 

3  3  4 
3  3  4 
3  3  4 

3  3  4 
3  4  4 
3  4  4 

.30 

.31 
.32 
.33 

.34 
.35 

.36 

.37 
.38 
.39 

1995 

2042 
2089 
2138 

2188 
2239 
2291 

2344 
2399 
2455 

2000 

2046 
2094 
2143 

2193 
2244 
2296 

2350 
2404 
2460 

2004 

2051 
2099 
2148 

2198 
2249 
2301 

2355 
2410 

2466 

2009 

2014 

2018 

2023 

2028 

2032 

2037 

oil 

2  2  3 

3  4  4 

2056 
2104 
2153 

2203 
2254 
2307 

2360 
2415 
2472 

2061 
2109 
2158 

2208 
2259 
2312 

2366 
2421 

2477 

2065 
2113 
2163 

2213 
2265 
2317 

2371 
2427 

2483 

2070 
2118 
2168 

2218 
2270 
2323 

2377 
2432 
2489 

2075 
2123 
2173 

2223 
2275 
2328 

2382 
2438 
2495 

2080 
2128 
2178 

2228 
2280 
2333 

2388 
2443 
2500 

2084 
2133 
2183 

2234 
2286 
2339 

2393 
2449 
2506 

oil 
oil 
oil 

112 
1  1  2 
1  1  2 

112 
1  1  2 
112 

2  2  3 
2  2  3 
2  2  3 

2  3  3 
2  3  3 
2  3  3 

2  3  3 
2  3  3 
2  3  3 

3  4  4 
3  4  4 

3  4  4 

4  4  5 
4  4  5 
4  4  5 

4  4  5 
4  5  5 

4  5  5 

.40 

2512 

2518 

2523 

2529 

2535 

2541 

2547 

2553 

2559 

2564 

112 

2  3  4 

4  5  5 

.41 
.42 
.43 

.44 
.45 

.46 

.47 

.48 
.49 

2570 
2630 
2692 

2754 

2818 
2884 

2951 
3020 
3090 

2576 
2636 
2698 

2761 

2825 
2891 

2958 
3027 
3097 

2582 
2642 
2704 

2767 
2831 
2897 

2965 
3034 
3105 

2588 
2649 
2710 

2773 

2838 
2904 

2972 
3041 
3112 

2594 
2655 
2716 

2780 
2844 
2911 

2979 
3048 
3119 

2600 
2661 
2723 

2786 
2851 
2917 

2985 
3055 
3126 

2606 
2667 
2729 

2793 
2858 
2924 

2992 
3062 
3133 

2612 
2673 
2735 

2799 
2864 
2931 

2999 
3069 
3141 

2618 
2679 

2742 

2805 
2871 
2938 

3006 
3076 
3148 

2624 
2685 
2748 

2812 
2877 
2944 

3013 
3083 
3155 

112 
1  1  2 
112 

112 
1  1  2 
1  1  2 

1  1  2 
112 

1  1  2 

2  3  4 
2  3  4 

2  3  4 

3  3  4 
3  3  4 
3  3  4 

3  3  4 
3  3  4 
3  4  4 

4  5  6 
4  5  6 
4  5  6* 

4  5  6 

5  5  6 
5  5  6 

5  6  6 
5  6  6 
5  6  6 

XIV] 

Table  XIV  &- 

-Antilogarithms  to  Four 

Places 

137 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

12  3 

4  5  6 

7  8  9 

.50 

3162 

3170 

3177 

3184 

3192 

3199 

3206 

3214 

3221 

3228 

112 

3  4  4 

5  6  7 

.51 

.52 
.53 

.54 
.55 
.56 

.57 
.58 
.59 

3236 
3311 

3388 

3467 
3548 
3631 

3715 
3802 
3890 

3243 
3319 
3396 

3475 
3556 
3639 

3724 
3811 
3899 

3251 
3327 
3404 

3483 
3565 
3648 

3733 
3819 
3VK)8 

3258 
3334 
3412 

3491 
3573 
3656 

3741 
3828 
3917 

3266 
3342 
3420 

3499 
3581 
3664 

3750 
3837 
3926 

3273 
3350 
3428 

3508 
3589 
3673 

3758 
3846 
3936 

3281 
3357 
3436 

3516 
3597 
3681 

3767 
3855 
3945 

3289 
3365 
3443 

3524 
3606 
3690 

3776 
3864 
3954 

3296 
3373 
3451 

a532 
3614 
3698 

3784 
3873 
3963 

3304 
3381 
3459 

3540 
3622 
3707 

3793 

3882 
3972 

1  1  2 
1  1  2 

1  2  2 

12  2 
1  2  2 
12  2 

12  3 
12  3 
12  3 

3  4  4 
3  4  5 
3  4  5 

3  4  5 
3  4  5 
3  4  5 

3  4  5 

3  4  5 

4  5  5 

5  6  7 

5  6  7 

6  6  7 

6  6  7 
6  7  7 
6  7  8 

6  7  8 
6  7  8 
6  7  8 

.60 

3981 

3990 

3999 

4009 

4018 

4027 

4036 

4046 

4055 

4064 

12  3 

4  5  6 

7  8  8 

.61 
.62 
.63 

.64 
.65 

.66 

.67 
.68 
.69 

4074 
4169 
4266 

4365 
4467 
4571 

4677 
4786 
4898 

4083 
4178 
4276 

4375 
4477 
4581 

4688 
4797 
4909 

4093 
4188 
4285 

4385 

4487 
4592 

4699 
4808 
4920 

4102 
4198 
4295 

4395 
4498 
4603 

4710 
4819 
4932 

4111 

4207 
4305 

4406 
4508 
4613 

4721 
4831 
4943 

4121 
4217 
4315 

4416 
4519 
4624 

4732 
4842 
4955 

4130 
4227 
4325 

4426 
4529 
4634 

4742 
4853 
4966 

4140 
4236 
4335 

4436 
4539 
4645 

4753 
4864 
4977 

4150 
4246 
4345 

4446 
4550 
4656 

4764 
4875 
4989 

4159 
4256 
4355 

4457 
45(i0 
4667 

4775 
4887 
5000 

12  3 
12  3 
12  3 

12  3 
1  2  3 
12  3 

12  3 

1  2  3 
1  2  3 

4  5  6 
4  5  6 
4  5  6 

4  5  6 
4  5  6 
4  5  6 

4  5  7 

5  6  7 
5  6  7 

7  8  9 
7  8  9 
7  8  9 

7  8  9 
7  8  9 

7  9  10 

8  9  10 
8  9  10 
8  910 

.70 

.71 

.72 
.73 

.74 
.75 
.76 

.77 
.78 
.79 

.80 

5012 

5023 

5035 

5047 

5058 

5070 

5082 

5093 

5105 

5117 

12  3 

5  6  7 

8  910 

5129 

5248 
5370 

5495 
5623 
5754 

588'8 
6026 
6166 

6310 

5140 
5260 
5383 

5508 
5636 
5768 

5902 
6039 
6180 

5152 
5272 
5395 

5521 
5649 

5781 

5916 
6053 
6194 

5164 
5284 
5408 

5534 
5662 
5794 

5929 
6067 
6209 

5176 
5297 
5420 

5546 
5675 
5808 

5943 
6081 
6223 

5188 
5309 
5433 

5559 
5689 
5821 

5957 
6095 
6237 

5200 
5321 
5445 

5572 
5702 
5834 

5970 
6109 
6252 

5212 
5333 
5458 

5585 
5715 
5848 

5984 
6124 
626() 

5224 
5346 
5470 

5598 
5728 
5861 

5908 
6138 
6281 

5236 
5358 
5483 

5610 
5741 
5875 

6012 
6152 
6295 

1  2  4 
1  2  4 
13  4 

1  3  4 
13  4 
1  3  4 

1  3  4 
1  3  4 
1  3  4 

5  6  7 
5  6  7 
5  6  7 

5  6  8 

5  7  8 
5  7  8 

5  7  8 

6  7  8 
6  7  9 

8  10  11 

9  10  11 
910  11 

910  12 
9  11  12 
9  11  12 

10  11  12 
10  11  13 
10  11  13 

6324 

6339 

6353 

6368 

6383 

6397 

6412 

6427 

6442 

13  4 

6  7  9 

10  12  13 

.81 
.82 
.83 

.84 
.85 

.86 

.87 
.88 
.89 

6457 
6607 
6761 

6918 
7079 
7244 

7413 
7586 
7762 

6471 
6622 
6776 

6934 
7096 
7261 

7430 
7603 
7780 

6486 
6637 
6792 

6950 
7112 
7278 

7447 
7621 

7798 

6501 
6653 
6808 

6966 
7129 
7295 

7464 

7638 
7816 

6516 
6668 
6823 

6982 
7145 
7311 

7482 
7656 
7834 

6531 
6683 
6839 

6998 
7161 
7328 

7499 
7674 

7852 

6546 
6699 
6855 

7015 
7178 
7345 

7516 
7691 
7870 

6561 
6714 

6871 

7031 
7194 
7362 

7534 
7709 
7889 

6577 
6730 

6887 

7047 
7211 
7379 

7551 

7727 
7907 

6592 
()745 
6902 

7063 
7228 
7396 

7568 
7745 
7925 

2  3  5 
2  3  5 
2  3  5 

2  3  5 
2  3  5 
2  3  5 

2  4  5 
2  4  5 
2  4  6 

6  8  9 
6  8  9 

6  8  9 

7  8  10 
7  8  10 
7  8  10 

7  910 
7  9n 
7  9  11 

11  12  14 
11  12  14 
11  13  14 

11  13  15 

12  13  15 
12  14  15 

12  14  16 

12  14  16 

13  15  K) 

.90 

7943 

7962 

7980 

7998 

8017 

8035 

8054 

8072 

8091 

8110 

2  4  6 

7  9  11 

13  15  17 

.91 
.92 
.93 

.94 
.95 

.96 

.97 
.98 
.99 

8128 
8318 
8511 

8710 
8913 
9120 

9333 
9550 
9772 

8147 
8337 
8531 

8730 
8933 
9141 

9354 
9572 
9795 

8166 
8356 
8551 

8750 
8954 
9162 

9376 
9594 
9817 

8185 
8375 
8570 

8770 
8974 
9183 

9397 
9616 
9840 

8204 
8395 
8590 

8790 
8995 
9204 

9419 
9638 
9863 

8222 
8414 
8610 

8810 
9016 
9226 

9441 
9661 
9886 

8241 
8433 
8630 

8831 
9036 
9247 

9462 
9683 
9908 

8260 
8453 
8650 

8851 
9057 
9268 

9484 
9705 
9931 

8279 
8472 
8670 

8872 
9078 
9290 

9506 
9727 
9954 

8299 
8492 
8690 

8892 
9099 
9311 

9528 
9750 
9977 

2  4  6 
2  4  6 
2  4  6 

2  4  6 
2  4  6 
2  4  6 

2  4  6 
2  4  7 
2  6  7 

8  9  11 
8  10  12 
8  10  12 

8  10  12 

8  10  12 

9  1113 

9  1113 
9  1113 
9  1114 

13  15  17 

14  15  17 
14  16  18 

14  16  18 

15  17  19 
15  17  19 

15  17  19 

16  18  20 
16  18  21 

138  Table  XIV  c  —  Four  Place  Trigonometric  Functions   [xiv 

[Characteristics  of  Logarithms  omitted  —  determine  by  the  usual  rule  from  the  value] 


Radtanr 

Degrees 

Sine 

Tangent 

Cotangent 

Cosine 

Xv.A.i-'X.a.jLi  o 

.M^  JltKx  3X*UM2iiy 

Value 

Lo^io 

Value 

Logio 

Value 

Logio 

Value 

Logio 

.0000 
.0029 

0°00' 

10 

.0000 

0000 

1.0000 

.0000 

9G°G0' 

50 

1.5708 
1.5679 

.0029 

.4637 

!0029 

.4637 

343.77 

.5363 

I'.OOOO 

!oooo 

.0058 

20 

.0058 

.7648 

.0058 

.7648 

171.89 

.2352 

1.0000 

.0000 

40 

1.5650 

.0087 

30 

.0087 

.9408 

.0087 

.9409 

114.59 

.0591 

1.0000 

.0000 

30 

1.5621 

.0116 

40 

.0116 

.0658 

.0116 

.0658 

85.940 

.9342 

.9999 

.0000 

20 

1.5592 

.0145 

50 

.0145 

.1627 

.0145 

.1627 

68.750 

.8373 

.9999 

.0000 

10 

1.5563 

.0175 

1°00' 

.0175 

.2419 

.0175 

.2419 

57.290 

.7581 

.9998 

.9999 

89°  GG' 

1.5533 

.0204 

10 

.0204 

.3088 

.0204 

.3089 

49.104 

.6911 

.9998 

.9999 

50 

1.5504 

.0233 

20 

.0233 

.3668 

.0233 

.3669 

42.964 

.6331 

.9997 

.9999 

40 

1.5475 

.0262 

30 

.0262 

.4179 

.0262 

.4181 

38.188 

.5819 

.9997 

.9999 

30 

1.5446 

.0291 

40 

.0291 

.4637 

.0291 

.4638 

34.368 

.5362 

.9996 

.9998 

20 

1.5417 

.0320 

50 

.0320 

.5050 

.0320 

.5053 

31.242 

.4947 

.9995 

.9998 

10 

1.5388 

.0349 

2°  00' 

.0349 

.5428 

.0349 

.5431 

28.636 

.4569 

.9994 

.9997 

88°  GG' 

1.5359 

.0378 

10 

.0378 

.5776 

.0378 

.5779 

26.432 

.4221 

.9993 

.9997 

50 

1.5330 

.0407 

20 

.0407 

.6097 

.0407 

.6101 

24.542 

.3899 

.9992 

.9996 

40 

1.5301 

.0436 

30 

.0436 

.6397 

.0437 

.6401 

22.904 

.3599 

.9990 

.9996 

30 

1.5272 

.0465 

40 

.0465 

.6677 

.0466 

.6682 

21.470 

.3318 

.9989 

.9995 

20 

1.5243 

.0495 

50 

.0494 

.6940 

.0495 

.6945 

20.206 

.3055 

.9988 

.9995 

10 

1.5213 

.0524 

3°  GO' 

.0523 

.7188 

.0524 

.7194 

19.081 

.2806 

.9986 

.9994 

87° GG' 

1.5184 

.0553 

10 

.0552 

.7423 

.0553 

.7429 

18.075 

.2571 

.9985 

.9993 

50 

1.5155 

.0582 

20 

.0581 

.7645 

.0582 

.7652 

17.169 

.2348 

.9983 

.9993 

40 

1.5126 

.0611 

30 

.0610 

.7857 

.0612 

.7865 

16.350 

.2135 

.9981 

.9992 

30 

1.5097 

.0640 

40 

.0640 

.8059 

.0641 

.8067 

15.605 

.1933 

.9980 

.9991 

20 

1.5068 

.0669 

50 

.0669 

.8251 

.0670 

.8261 

14.924 

.1739 

.9978 

.9990 

10 

1.5039 

.0698 

4°  00' 

.0698 

.8436 

.0699 

.8446 

14.301 

.1554 

.9976 

.9989 

86°  GG' 

1.5010 

.0727 

10 

.0727 

.8613 

.0729 

.8624 

13.727 

.1376 

.9974 

.9989 

50 

1.4981 

.0756 

20 

.0756 

.8783 

.0758 

.8795 

13.197 

.1205 

.9971 

.9988 

40 

1.4952 

.0785 

30 

.0785 

.8946 

.0787 

.8960 

12.706 

.1040 

.9969 

.9987 

30 

1.4923 

.0814 

40 

.0814 

.9104 

.0816 

.9118 

12.251 

.0882 

.9967 

.9986 

20 

1.4893 

.0844 

50 

.0843 

.9256 

.0846 

.9272 

11.826 

.0728 

.9964 

.9985 

10 

1.4864 

.0873 

5°  GO' 

.0872 

.9403 

.0875 

.9420 

11.430 

.0580 

.9962 

.9983 

85°  GG' 

1.4835 

.0902 

10 

.0901 

.9545 

.0904 

.9563 

11.059 

.0437 

.9959 

.9982 

50 

1.4806 

.0931 

20 

.0929 

.f)682 

.0934 

.9701 

10.712 

.0299 

.9957 

.9981 

40 

1.4777 

.0960 

30 

.0958 

.9816 

.0963 

.9836 

10.385 

.0164 

.9954 

.9980 

30 

1.4748 

.0989 

40 

.0987 

.9945 

.0992 

.9966 

10.078 

.0034 

.9951 

.9979 

20 

1.4719 

.1018 

50 

.1016 

.0070 

.1022 

.0093 

9.7882 

.9907 

.9948 

.9977 

10 

1.4690 

.1047 

6°GG' 

.1045 

.0192 

.1051 

.0216 

9.5144 

.9784 

.9945 

.9976 

84° GG' 

1.4661 

.1076 

10 

.1074 

.0311 

.1080 

.0336 

9.2553 

.9664 

.9942 

.9975 

50 

1.4632 

.1105 

20 

.1103 

.0426 

.1110 

.0453 

9.0098 

.9547 

.9939 

.9973 

40 

1.4603 

.1134 

30 

.1132 

.0539 

.1139 

.0567 

8.7769 

.9433 

.9936 

.9972 

30 

1.4573 

.1164 

40 

.1161 

.0648 

.1169 

.0678 

8.5555 

.9322 

.9932 

.9971 

20 

1.4544 

.1193 

50 

.1190 

.0755 

.1198 

.0786 

8.3450 

.9214 

.9929 

.9969 

10 

1.4515 

.1222 

7°  GO' 

.1219 

.0859 

.1228 

.0891 

8.1443 

.9109 

.9925 

.9968 

83° GG' 

1.4486 

.1251 

10 

.1248 

.0961 

.1257 

.0995 

7.9530 

.9005 

.9922 

.9966 

^0 

1.4457 

.1280 

20 

.1276 

.1060 

.1287 

.109() 

7.7704 

.8904 

.9918 

.9964 

lo 

1.4428 

.1309 

30 

.1305 

.1157 

.1317 

.1194 

7.5958 

.8806 

.9914 

.9963 

30 

1.4399 

.1338 

40 

.1334 

.1252 

.1346 

.1291 

7.4287 

.8709 

.9911 

.9961 

20 

1.4370 

.1367 

50 

.1363 

.1345 

.1376 

.1385 

7.2687 

.8615 

.9907 

.9959 

10 

1.4341 

.1396 

8°GG' 

.1392 

.1436 

.1405 

.1478 

7.1154 

.8522 

.9903 

.9958 

82°  GG' 

1.4312 

.1425 

10 

.1421 

.1525 

.1435 

.1569 

6.9682 

.8431 

.9899 

.9956 

50 

1.4283 

.1454 

20 

.1449 

.1612 

.1465 

.1658 

6.8269 

.8342 

.9894 

.9954 

40 

1.4f^54 
l.i224 

.148^1 

30 

.1478 

.1697 

.1495 

.1745 

6.6912 

.8255 

.9890 

.9952 

30 

.1513 

40 

.1507 

.1781 

.1524 

.1831 

6.5606 

.8169 

.9886 

.9950 

20 

1.4195 

.1542 

50 

.1536 

.1863 

.1554 

.1915 

6.4348 

.8085 

.9881 

.9948 

10 

1.4166 

.1571 

9°GG' 

.1564 

.1943 

.1584 

.1997 

6.3138 

.8003 

.9877 

.9946 

81°  GG' 

1.4137 

Value 

Logio 

Value 

Login 

Value 

Logio 

Value 

Logio 

Degbbes 

Radians 

Cosine 

Cotangent 

Tangent 

Sine 

XIV] 


Four  Place  Trigonometric  Functions 


[Characteristics  of  Log-arith 

ms  omitted  —  determine  by  the  usual  rule  from  the  value] 

T?  A  T»T  A  VR 

Dkg-rees 

Sine 

Tangent 

Cotangent 

Cosine 

XW.d.l^i>-^^.i.^  O 

1^  Xm  \T  Xk  XU  A.' 0 

S^alue 

Logio 

Value 

Loi^io 

Value 

Lo?io 

Value 

I^Offio 

.1571 

9°  00 

.1564 

.1^)43 

.15^ 

.1997 

6.3138 

.8003 

.9877 

.91>4() 

81°  00' 

1.4137 

.1600 

10 

.1593 

.2022 

.1614 

.2078 

6.1970 

.7922 

.9872 

.9944 

50 

1.4108 

.1029 

20 

.1622 

.2100 

.1644 

.2158 

6.0844 

.7842 

.9868 

.9942 

40 

1.4079 

.1658 

30 

.1650 

.217() 

.1673 

.2236 

5.9758 

.7764 

.98()3 

.9940 

30 

1.4050 

.1687 

40 

.1679 

.2251 

.1703 

.2313 

5.8708 

.7687 

.9858 

.9938 

20 

1.4021 

.1716 

50 

.1708 

.2324 

.1733 

.2389 

5.76V)4 

.7611 

.9853 

.9^):36 

10 

1.3992 

.1745 

10^00' 

.1736 

.2397 

.1763 

.2463 

5.6713 

.7537 

.9848 

.9934 

80°  00' 

1..3963 

.1774 

10 

.1765 

.2468 

.1793 

.2536 

5.5764 

.7464 

.9843 

.9931 

50 

1.3934 

.1804 

20 

.1794 

.2538 

.1823 

.2609 

5.4845 

.7391 

.9838 

.9929 

40 

1.3904 

.1833 

30 

.1822 

.260<3 

.1853 

.2680 

5.3955 

.7320 

.9833 

.9927 

30 

1.3875 

.1862 

40 

.1851 

.2674 

.1883 

.2750 

5.3093 

.7250 

.9827 

.9924 

20 

1.3846 

.1891 

50 

.1880 

.2740 

.1914 

.2819 

5.2257 

.7181 

.9822 

.9922 

10 

1.3817 

.1920 

11°00' 

.1^)08 

.2806 

.1944 

.2887 

5.1446 

.7113 

.9816 

.9919 

79°  00' 

1.3788 

.1949 

10 

.1937 

.2870 

.1974 

.2953 

5.0658 

.7047 

.9811 

.9917 

50 

1.3759 

.1978 

20 

.1965 

.2934 

.2004 

.3020 

4.98i)4 

.6980 

.9805 

.9914 

40 

1.3730 

.2007 

30 

.1994 

.2997 

.2035 

.3085 

4.9152 

.6915 

.9799 

.9<)12 

30 

1.3701 

.2036 

40 

.2022 

.3058 

.20(>5 

.3149 

4.8430 

.6851 

.9793 

.9909 

20 

1.3672 

.2065 

50 

.2051 

.3119 

.2095 

.3212 

4.7729 

.6788 

.9787 

.9907 

10 

1.3&43 

.2094 

12°  00' 

.2079 

.3179 

.2126 

.3275 

4.7046 

.6725 

.9781 

.9904 

78°  00' 

1.3614 

.2123 

10 

.2108 

.3238 

.2156 

.3336 

4.6382 

.6664 

.9775 

.9i)01 

50 

1.3584 

.21.53 

20 

.2136 

.3296 

.2186 

.3397 

4.5736 

.6603 

.9769 

.9899 

40 

1.3555 

.2182 

30 

.21f>4 

.3353 

.2217 

.^3458 

4.5107 

.6542 

.9763 

.9896 

30 

1.3526 

.2211 

40 

.2193 

.3410 

.2247 

.3517 

4.4494 

.6483 

.9757 

.9893 

20 

1.3497 

.2240 

50 

.2221 

.3466 

.2278 

.3576 

4.3897 

.6424 

.9750 

.9890 

10 

1.3468 

.2269 

13°  00' 

.2250 

.3521 

.2309 

.3634 

4.3315 

.6366 

.9744 

.9887 

77° 00' 

1.3439 

.2298 

10 

.2278 

.3575 

.2339 

.3691 

4.2747 

.6309 

.9737 

.9884 

50 

IMIO 

.2327 

20 

.2306 

.3629 

.2370 

.3748 

4.2193 

.6252 

.9730 

.9881 

40 

1.3381 

.2356 

30 

.23;^ 

.3682 

.2401 

.3804 

4.1653 

.6196 

.9724 

.9878 

30 

l.e3352 

.2385 

40 

.2:3(53 

.3734 

.2432 

.3859 

4.1126 

.6141 

.9717 

.9875 

20 

1.3323 

.2414 

50 

.2391 

.3786 

.2462 

.3914 

4.0611 

.6086 

.9710 

.9872 

10 

1.3294 

.2443 

14°  00' 

.2419 

.3837 

.2493 

.3968 

4.0108 

.6032 

.9703 

.9869 

76°  00' 

1.3265 

.2473 

10 

.2447 

.3887 

.2524 

.4021 

3.9fn7 

.5979 

.9696 

.9866 

50 

1.3235 

.2502 

20 

.2476 

.3937 

.2555 

.4074 

3.9136 

.5926 

.9689 

.9863 

40 

1.3206 

.2531 

30 

.2504 

.3986 

.2586 

.4127 

3.8667 

.5873 

.9681 

.9859 

30 

1.3177 

.2;560 

40 

.2532 

.4035 

.2617 

.4178 

3.8208 

.5822 

.9674 

.985() 

20 

1.3148 

.2589 

50 

.2560 

.4083 

.2648 

.4230 

3.7760 

.5770 

.9(367 

.9853 

10 

1.3119 

.2618 

15°00' 

.2588 

.4130 

.2679 

.4281 

3.7321 

.5719 

.9659 

.9849 

75° 00' 

1.3090 

.2647 

10 

.2616 

.4177 

.2711 

.4331 

3.6891 

.5669 

.9652 

.984(5 

50 

1.3061 

.2676 

20 

.2644 

.4223 

.2742 

.4381 

3.6470 

.5619 

.9644 

.9843 

40 

1.3032 

.2705 

30 

.2672 

.4269 

.2773 

.4430 

3.6059 

.5570 

.9{)36 

.9839 

30 

1.3003 

.27:34 

40 

.2700 

.4314 

.2805 

.4479 

3.5656 

.5521 

.9628 

.9836 

20 

1.2974 

.2763 

50 

.2728 

.4359 

.2836 

.4527 

3.5261 

.5473 

.9621 

.9832 

10 

1.2945 

,2793 

16°  00' 

.2756 

.4403 

.2867 

.4575 

3.4874 

.5425 

.9613 

.9828 

74° 00' 

1.2915 

.2822 

10 

.2784 

.4447 

.2899 

.4622 

3.4495 

.5378 

.9605 

.9825 

50 

1.2886 

.2851 

20 

.2812 

.4491 

.2931 

.4669 

3.4124 

.5331 

.9596 

.9821 

40 

1.2857 

.2880 

30 

.2840 

.4533 

.2962 

.4716 

3.3759 

.5284 

.9588 

.9817 

30 

1.2828 

.2()09 

40 

.2868 

.4576 

.2994 

.4762 

3.3402 

.5238 

.9580 

.9814 

20 

1.2799 

.2938 

50 

.2896 

.4618 

.3026 

.4808 

3.3052 

.5192 

.9572 

.9810 

10 

1.2770 

.2f)67 

17°  00' 

.2924 

.4659 

.3057 

.4^53 

3.2709 

.5147 

.9563 

.9806 

73°  00' 

1.2741 

.29m 

10 

.2952 

.4700 

.3089 

.4898 

3.2371 

.5102 

.9555 

.9802 

50 

1.2712 

,  m5 

20 

.2979 

.4741 

.3121 

.4^)43 

3.2041 

.5057 

.9546 

.9798 

40 

1.2683 

.3054 

30 

.3007 

.4781  I  .3153 

.4987 

3.1716 

.5013 

.9537 

.9794 

30 

1.2654 

.3083 

40 

.3035 

.4821 

.3185 

.5031 

3.1397 

.4969 

.9528 

.9790 

20 

1.2625 

.3113 

50 

.3062 

.4861 

.3217 

.5075 

3.1084 

.4925 

.9520 

.9786 

10 

1.2595 

.3142 

18° 00' 

.3090 

.4900 

.3249 

.5118 

3.0777 

.4882 

.9511 

.9782 

72°  00' 

1.2566 

Value 

Logio 

Value 

Logio 

Value 

Lo^io 

Value 

Logio 

Degrees 

Radians 

COSIXE 

Cotangent 

Tangent 

Sine 

140  Four  Place  Trigonometric  Functions  [xiv 

[Characteristics  of  Logarithms  omitted  —  determine  by  the  usual  rule  from  the  value] 


RADIAJNfi 

Degrees 

Sine 

Tangent 

Cotangent  |   Cosine 

^%)JX.X/±f^~i.^  o 

Value 

Logio 

Value 

Logio 

Value 

Logio  Value 

Logio 

.3142 

18° 00' 

.3090 

.4900 

.3249 

.5118 

3.0777 

.4882  i  .9511 

.9782 

72°  00' 

1.2566 

.3171 

10 

.3118 

.4939 

.3281 

.5161 

3.0475 

.4839  i  .9502 

.9778 

50 

1.2537 

.3200 

20 

.3145 

.4977 

.3314 

.5203 

3.0178 

.4797  .9492 

.9774 

40 

1.2508 

.3229 

30 

.3173 

.5015 

.3346 

.5245 

2.9887 

.4755  .9483 

.9770 

30 

1.2479 

.3258 

40 

.3201 

.5052 

.3378 

.5287 

2.9600 

.4713  .9474 

.9765 

20 

1.2450 

.3287 

50 

.3228 

.50^)0 

.;3411 

.5329 

2.9319 

.4671  I  .9465 

.9761 

10 

1.2421 

.3316 

19°  00' 

.3256 

.5126 

.3443 

.5370 

2.f)042 

.4630 

.9455 

.9757 

71°  00' 

1.2392 

.3345 

10 

.3283 

.5163 

.3476 

.5411 

2.8770 

.4589 

.9446 

.9752 

50 

1.2363 

.3374 

20 

.3311 

.5199 

.3508 

.5451 

2.8.502 

.4549 

.9436 

.9748 

40 

1.2334 

.3403 

30 

.3338 

.5235 

.3541 

.5491 

2.8239 

.4509 

.9426 

.9743 

30 

1.2:305 

.3432 

40 

.3365 

.5270 

.3574 

.5531 

2.7980 

.4469 

.9417 

.9739 

20 

1.2275 

.3462 

50 

.3393 

.5;306 

.3607 

.5571 

2.7725 

.4429 

.9407 

.9734 

10 

1.2246 

.3491 

20°  00' 

.3420 

..5341 

.3640 

.5611 

2.7475 

.4389 

.9397 

.9730 

70°  00' 

1.2217 

.3520 

10 

.3448 

.5375 

.3673 

.5650 

2.7228 

.43,50 

.9387 

.9725 

50 

1.2188 

.3549 

20 

.3475 

.5409 

.'Sim 

.5689 

2.6985 

.4311 

.9377 

.9721 

40 

1.2159 

.3578 

30 

.3502 

.5443 

.3739 

.5727 

2.6746 

.4273 

.9367 

.9716 

30 

1.2130 

.3607 

40 

.3529 

.5477 

.3772 

.5766 

2.6511 

.42:34 

.9356 

.9711 

20 

1.2101 

.3636 

50 

.3557 

.5510 

.3805 

.5804 

2.6279 

.4196 

.9346 

.9706 

10 

1.2072 

.3665 

21°  00' 

.3584 

.5543 

.3839 

.5842 

2.6051 

.4158 

.9336 

,9702 

69° 00' 

1.2043 

.3694 

10 

.3611 

.5576 

.3872 

.5879 

2.5826 

.4121 

.9325 

.9697 

50 

1.2014 

.3723 

20 

.3638 

.5609 

.3906 

.5917 

2.5605 

.4083 

.9315 

.9692 

40 

1.1985 

.3752 

30 

.3665 

.5641 

.3939 

.5954 

2.5386 

.4046 

.9304 

.9687 

30 

1.1956 

.3782 

40 

.3692 

.5673 

.3973 

.5f)91 

2.5172 

.4009 

.9293 

.9682 

20 

1.1926 

.3811 

50 

.3719 

.5704 

.4006 

.6028 

2.4960 

.3972 

.9283 

.9677 

10 

1.1897 

.3840 

22° 00' 

.3746 

.5736 

.4040 

.6064 

2.4751 

.3936 

.9272 

.9672 

68° 00' 

1.1868 

.:3869 

10 

.3773 

.5767 

.4074 

.6100 

2.4545 

.3900 

.9261 

.9667 

50 

1.1839 

.3898 

20 

.3800 

.5798 

.4108 

.6136 

2.4342 

.3864 

.9250 

.9661 

40 

1.1810 

.3927 

30 

.3827 

.5828 

.4142 

.6172 

2.4142 

.3828 

.9239 

.9656 

30 

1.1781 

.3956 

40 

.3854 

.5859 

.4176 

.6208 

2.3945 

.3792 

.9228 

.9651 

20 

1.1752 

.3985 

50 

.3881 

.5889 

.4210 

.6243 

2.3750 

.3757 

.9216 

.9646 

10 

1.1723 

.4014 

23° 00' 

.3907 

.5919 

.4245 

.6279 

2.3559 

.3721 

.9205 

.9640 

67° 00' 

1.1694 

.4043 

10 

.39:^ 

.5948 

.4279 

.6314 

2.3369 

.3686 

.9194 

.9635 

50 

1.1665 

.4072 

20 

.3%1 

.5978 

.4314 

.6348 

2.3183 

.3652 

.9182 

.9629 

40 

1.1636 

.4102 

30 

.3987 

.6007 

.4348 

.6383 

2.2998 

.3617 

.9171 

.9624 

30 

1.1606 

.4131 

40 

.4014 

.mm 

.4383 

.6417 

2.2817 

.3583 

.9159 

.9618 

20 

1.1577 

.4160 

50 

.4041 

.6065 

.4417 

.6452 

2.2637 

.3548 

.9147 

.9613 

10 

1.1548 

.4189 

24° 00' 

.4067 

.6093 

.4452 

.6486 

2.'L.m 

.3514 

.9135 

.9607 

66° 00' 

1.1519 

.4218 

10 

.4094 

.6121 

.4487 

.6520 

-2  2286 

.3480 

.9124 

.9602 

50 

1.1490 

.4247 

20 

.4120 

.6149 

.4522 

.6553 

2.2113 

.3447 

.9112 

.9596 

40 

1.1461 

.4276 

30 

.4147 

.6177 

.4557 

.6587 

2.1913 

.3413 

.9100 

.P59() 

30 

1.1432 

.4305 

40 

.4173 

.6205 

.4592 

.6620 

2.1775 

.3380 

.9088 

.9584 

20 

1.1403 

.4334 

50 

.4200 

.6232 

.4628 

.6654 

2.1609 

.3346 

.9075 

.9579 

10 

1.1374 

.4363 

26° 00' 

.4226, 

.6259 

.4663 

.6687 

2.1445 

.3313 

.9063 

.9573 

65° 00' 

1.1345 

.4392 

10 

.4253 

.6286 

.4699 

.6720 

2.1283 

.3280 

.9051 

.9567 

50 

1.1:316 

.4422 

20 

.4279 

.6313 

.47*^ 

.6752 

2.1123 

.3248 

.9038 

.9561 

40 

1.1286 

.4451 

30 

.4305 

.6340 

.4770 

.6785 

2.0965 

.3215 

.9026 

.9555 

30 

1.1257 

.4480 

40 

.4331 

.6366 

.4806 

.6817 

2.0809 

.3183 

.9013 

.9549 

20 

1.1228 

.4509 

50 

.4358 

.6392 

.4841 

.6850 

2.0655 

.3150 

.9001 

.9543 

10 

1.1199 

.4538 

26° 00' 

.4384 

.6418 

.4877 

.6882 

2.0503 

.3118 

.8988 

.9537 

64°  00' 

1.1170 

.4567 

10 

.4410 

.6444 

.4913 

.6914 

2.0a53 

.3086 

.8975 

.9530 

50 

1.1141 

.4596 

20 

.4436 

.6470 

.4950 

.6946 

2.0204 

.3054 

.8962 

.9524 

40 

1.1112 

.4625 

30 

.4462 

.6495 

.4986 

.f]977 

2.0057 

.3023 

.8949 

.9518 

30 

1.1083 

.4654 

40 

.4488 

.6521 

.5022 

.7009 

1.9912 

.2991 

.8936 

.9512 

20 

1.1054 

.4683 

50 

.4514 

.6546 

.5059 

.7040 

1.9768 

.2960 

.8923 

.9505 

10 

1.1025 

.4712 

27°  00' 

.4540 

.6570 

.5095 

.7072 

1.9626 

.2928 

.8910 

.9499 

63°  00' 

1.0996 

Value 

Logio 

Value 

Logio 

Value 

Logio 

Value 

Logio 

Degrees 

Radians 

Cosine 

Cotangent 

Tangent   | 

Sine    | 

XIV]  Four  Place  Trigonometric  Functions 

[Characteristics  of  Logarithms  omitted  —  determine  by  the  usual  rule  from  the  vahie] 


Radians 

Degrees 

Sine 

Tangent 

Cotangent 

Cosine 

Value 

Logio 

Value 

Logio 

Value 

Logio 

Value  Logio 

.4712 

27°  00' 

.4540 

.6570 

.5095 

.7072 

1.9626 

.2928 

.8910  .9499 

63° 00' 

1.091^ 

.4741 

10 

.45(^ 

.6595 

.5132 

.7103 

1.9486 

.2897 

.8897  .9492 

50 

1.09(J6 

.4771 

20 

.4592 

.6620 

.5169 

.7134 

1.9347 

.2866 

.8884  .9486 

40 

1.0937 

.4800 

30 

.4617 

.6644 

.5206 

.7165 

1.9210 

.2835 

.8870  .9479 

30 

1.0908 

.4829 

40 

.4643 

.6668 

.5243 

.7196 

1.9074 

.2804 

.8857  .9473 

20 

1.0879 

.4858 

50 

.4669 

.6692 

.5280 

.7226 

1.8940 

.2774 

.8843  .9466 

10 

1.0850 

.4887 

28°  00' 

.4695 

.6716 

.5317 

.7257 

1.8807 

.2743 

.8829  .9459 

62°  00' 

1.0821 

.4916 

10 

.4720 

.6740 

.5354 

.7287 

1.8676 

.2713 

.8816  .9453 

50 

1.0792 

AM5 

20 

.4746 

.6763 

.5392 

.7317 

1.8546 

.2683 

.8802  .9446 

40 

1.0763 

.4974 

30 

.4772 

.6787 

.5430 

.7348 

1.8418 

.2652 

.8788  .9439 

30 

1.0734 

.5003 

40 

.4797 

.6810 

.5467 

.7378 

1.8291 

.2(^22 

.8774  .9432 

20 

1.0705 

.5032 

50 

.4823 

.6833 

.5505 

.7408 

1.8165 

.2592 

.8760  .9425 

10 

1.0676 

.5061 

29° 00' 

.4848 

.6856 

.5543 

.7438 

1.8040 

.2562 

.8746  .9418 

61°  00' 

1.0647 

..5091 

10 

.4874 

.6878 

.5581 

.7467 

1.7917 

.2533 

.8732  .9411 

50 

1.0617 

.5120 

20 

.4899 

.6901 

.5619 

.7497 

1.7796 

.2503 

.8718  .9404 

40 

1.0588 

.5149 

30 

.4924 

.6923 

.5658 

.7526 

1.7675 

.2474 

.8704  .9397 

30 

1.0559 

.5178 

40 

.4950 

.6946 

.5696 

.7556 

1.7556 

.2444 

.8689  .9390 

20 

1.0530 

.5207 

50 

.4975 

.6968 

.5735 

.7585 

1.7437 

.2415 

.8675  .9383 

10 

1.0501 

.5236 

30°  00' 

.5000 

.6990 

.5774 

.7614 

1.7321 

.2386 

.8660  .9375 

60° 00' 

1.0472 

.5265 

10 

.5025 

.7012 

.5812 

.7644 

1.7205 

.2356 

.8646  .9368 

50 

1.0443 

.5294 

20 

.5050 

.7033 

.5851 

.7673 

1.7090 

.2327  i  .8631  .9361 

40 

1.0414 

.5323 

30 

.5075 

.7055 

.58^K) 

.7701 

1.6977 

.2299  :  .8616  .9353 

30 

1.0385 

.5352 

40 

.5100 

.7076 

.5930 

.7730 

1.6864 

.2270 

.8f>01  .9346 

20 

1.0356 

.5381 

50 

.5125 

.7097 

.5969 

.7759 

1.6753 

.2241 

.8587  .9338 

10 

1.0327 

.5411 

31° 00' 

.5150 

.7118 

.6009 

.7788 

1.6643 

.2212 

.8572  .9331 

59° 00' 

1.0297 

.5440 

10 

.5175 

.7139 

.6048 

.7816 

1.6534 

.2184  :  .8557  .9323 

50 

1.0268 

.5469 

20 

.5200 

.7160 

.6088 

.7845 

1.6426 

.2155  !  .8542  .9315 

40 

1.0239 

.5498 

30 

.5225 

.7181 

.6128 

.7873 

1.6319 

.2127  1  .8526  .9308 

30 

1.0210 

.5527 

40 

.5250 

.7201 

.6168 

.7902 

1.6212 

.2098 

.8511  .9300 

20 

1.0181 

.5556 

50 

.5275 

.7222 

.6208 

.7930 

1.6107 

.2070 

.8496  .9292 

10 

1.0152 

.5585 

32° 00' 

.5299 

.7242 

.6249 

.7958 

1.6003 

.2042 

.8480  .9284 

58°  00' 

1.0123 

.5614 

10 

.5324 

.7262 

.6289 

.7986 

1.5900 

.2014 

.8465  .9276 

50 

1.0094 

.5643 

20 

.5348 

.7282 

.6330 

.8014 

1.5798 

.1986 

.8450  .9268 

40 

1.0065 

.5672 

30 

.5373 

.7302 

.6371 

.8042 

1.5697 

.1958 

.8434  .9260 

30 

1.0036 

.5701 

40 

.5398 

.7322 

.6412 

.8070 

1.5597 

.1930 

.8418  .9252 

20 

1.0007 

.5730 

50 

.5422 

.7342 

.6453 

.8097 

1.5497 

.1903 

.8403  .9244 

10 

.9977 

.5760 

33° 00' 

.5446 

.7361 

.6494 

.8125 

1.5399 

.1875 

.8387  .9236 

57°  00' 

.9948 

.5789 

10 

.5471 

.7380 

.6536 

.8153 

1.5301 

.1847  !  .8371  .9228 

50 

.9^)19 

.5818 

20 

.5495 

.7400 

.6577 

.8180 

1.5204 

.1820  !  .8355  .9219 

40 

.9890 

.5847 

30 

.5519 

.7419 

.6619 

.8208 

1.5108 

.1792  ;  .8339  .9211 

30 

.9861 

.5876 

40 

.5544 

.7438 

.6()61 

.8235 

1.5013 

.1765  ;  .8323  .9203 

20 

.9832 

.5905 

50 

.5568 

.7457 

.6703 

.8263 

1.4919 

.1737  .8307  .9194 

10 

.9803 

.5934 

34° 00' 

.5.592 

.7476 

.6745 

.8290 

1.4826 

.1710  .8290  .9186 

56°  00' 

.9774 

.5963 

10 

.5(516 

.7494 

.()787 

.8317 

1.4733 

.1083 !  .8274  .9177 

50 

.9745 

.5992 

20 

.5640 

.7513 

.6830 

.8344 

1.4641 

.1656!  .8258^  .9169 

40 

.9716 

.6021 

30 

.5664 

.7531 

.6873 

.8371 

1.4550 

.16291  .8241  .9160 

30 

.9687 

.6050 

40 

.5688 

.7550 

.6916 

.8398 

1.4460 

.1602 

.8225  .9151 

20 

.9657 

.6080 

50 

.5712 

.7568 

.6959 

.8425 

1.4370 

.1575 

.8208  .9142 

10 

.9628 

.6109 

35° 00' 

.5736 

.7586 

.7002 

.8452 

1.4281 

.1548 

.8192  .9134 

55° 00' 

.9599 

.6138 

10 

.5760 

.7604 

.7046 

.8479 

1.4193 

.1521  j  .8175  .9125 

50 

.9570 

.6167 

20 

.5783 

.7622 

.7089 

.850<5 

1.4106 

.1494  ;  .8158  .911() 

40 

.9541 

.6196 

30 

.5807 

.7640 

.7133 

.8533 

1.4019 

.1467 

.8141  .9107 

30 

.9512 

.6225 

40 

.5831 

.7657 

.7177 

.8559 

1.39ri4 

.1441 

.8124  .9098 

20 

.9483 

.6254 

50 

.5854 

.7675 

.7221 

.8586 

1.3848 

.1414 

.8107  .9089 

10 

.9454 

.6283 

36° 00' 

.5878 

.7692 

.7265 

.8613 

1.3764 

.1387 

.8090  .9080 

54°  00' 

.9425 

Value 

Logio 

Value 

Logio 

Value 

Logio 

Value   Logio 

Degrees 

Radians 

Cosine 

Cotangent  |   Tangent 

Sine 

142  Four  Place  Trigonometric  Functions  [xiv 

[Characteristics  of  Logarithms  omitted  —  determine  by  the  usual  rule  from  the  valuej 


Radians 

Degrees 

Sine 

Tangent 

Cotangent 

Cosine 

^alue  Logio 

Value   Logic 

Value   Logio 

Value  Logio 

.6283 

36^00' 

.5878  .7692 

.7265  .8613 

1.3764  .1387 

.8090  .9080 

54° 00' 

.9425 

.63]  2 

10 

.5901  .7710 

.7310  .8639 

1.3680  .1361 

.8073  .9070 

50 

.9396 

.6341 

20 

.5925  .7727 

.7355  .8666 

1.3597  .1334 

.8056  smi 

40 

.9367 

.6370 

30 

.5948  .7744 

.7400  .8692 

1..3514  .1308 

.8039  .9052 

30 

.9338 

.6400 

40 

.5972  .7761 

.7445  .8718 

1.3432  .1282 

.8021  .9042 

20 

.9308 

.6429 

50 

.5995  .7778 

.7490  .8745 

1.3351  .1255 

.8004  .9033 

10 

.9279 

.6458 

37° 00' 

.6018  .7795 

.75.36  .8771 

1.3270  .1229 

.7986  .9023 

53°  00' 

.9250 

.6487 

10 

.6041  .7811 

.7581  .8797 

1.3190  .1203 

.7969  .9014 

50 

.9221 

.6516 

20 

.6065  .7828 

.7627  .8824 

1.3111  .1176 

.7951  .9004 

40 

.9192 

.6545 

30 

.6088  .7844 

.7673  .8850 

1.3032  .1150 

.7934  .8995 

30 

.9163 

.6574 

40 

.6111  .7861 

.7720  .8876 

1.2954  .1124 

.7916  .8985 

20 

.9134 

.6603 

50 

.6134  .7877 

.7766  .8902 

1.2876  .1098 

.7898  .8975 

10 

.9105 

.6632 

38°  00' 

.6157  .7893 

.7813  .8928 

1.2799  .1072 

.7880  .8^)65 

52°  00' 

.9076 

.6661 

10 

.6180  .7910 

.7860  .8954 

1.2723  .1046 

.7862  .8955 

50 

.9047 

.6690 

20 

.6202  .7926 

.7907  .8980 

1.2647  .1020 

.7844  .8945 

40 

.9018 

.6720 

30 

.6225  .7941 

.7954  Sm6 

1.2572  .0994 

.7826  .8935 

■SO 

.8988 

.6749 

40 

.6248  .7957 

.8002  .9032 

1.2497  .0968 

.7808  .8925 

20 

.8959 

.6778 

50 

.6271  .7973 

.8050  .9058 

1.2423  .0942 

.7790  .8915 

10 

.8930 

.6807 

39°  00' 

.6293  .7989 

.8098  .9084 

1.2349  .0916 

.7771  .8905 

51°00' 

.8901 

.68.36 

10 

.631(5  .8004 

.8146  .9110 

1.2276  .0890 

.7753  .8895 

50 

.8872 

.6865 

20 

.6338  .8020 

.8195  .9135 

1.2203  .0865 

.7735  .8884 

40 

.8843 

.6894 

30 

.6.361  .8035 

.8243  .9161 

1.2131  .0839 

.7716  .8874 

30 

.8814 

.6923 

40 

.6383  .8050 

.8292  .9187 

1.2059  .0813 

.7698  .8864 

20 

.8785 

.6952 

50 

.6406  .8066 

.8342  .9212 

1.1988  .0788 

.7679  .8853 

10 

.8756 

.6981 

40°  00' 

.6428  .8081 

.8391  .9238 

1.1918  .0762 

.7660  .8843 

50° 00' 

.8727 

.7010 

10 

.6450  .8096 

.8441  .9264 

1.1847  .0736 

.7642  .8832 

50 

.8698 

.7039 

20 

.6472  .8111 

.8491  .9289 

1.1778  .0711 

.7623  .8821 

40 

.8668 

.7069 

30 

.6494  .8125 

.8541  .9315 

1.1708  .0685 

.7604  .8810 

30 

.8639 

.7098 

40 

.6517  .8140 

.8591  .9.341 

1.1640  .0659 

.7585  .8800 

20 

.8610 

.7127 

50 

.6539  .8155 

.8642  .9366 

1.1571  .06.34 

.7566  .8789 

10 

.8581 

.7156 

41°00' 

.6561  .8169 

.8693  .9392 

1.1504  .0608 

.7547  .8778 

49°  00' 

.8552 

.7185 

10 

.6583  .8184 

.8744  .9417 

1.1436  .0583 

.7528  .8767 

50 

.8523 

.7214 

20 

.6604  .8198 

.8796  .9443 

1.1369  .0557 

.7509  .8756 

40 

.8494 

.7243 

30 

.6626  .8213 

.8847  .9468 

1.1303  .0532 

.7490  .8745 

30 

.8465 

.7272 

40 

.6648  .8227 

.8899  .9494 

1.1237  .0506 

.7470  .8733 

20 

.8436 

.7301 

50 

.6670  .8241 

.8952  .9519 

1.1171  .0481 

.7451  .8722 

10 

.8407 

.7330 

42°  00' 

.6691  .8255 

.9001  .9544 

1.1106  .0456 

.7431  .8711 

48°  00' 

.8378 

.7359 

10 

.6713  .8269 

.9057  .9570 

1.1041  .0430 

.7412  .8699 

50 

.8348 

.7389 

20 

.6734  .8283 

.9110  .9595 

1.0977  .0405 

.7392  .8688 

40 

.8319 

.7418 

30 

.6756  .8297 

.9163  .9621 

1.0913  .0379 

.7373  .8676 

30 

.8290 

.7447 

40 

.6777  .8311 

.9217  .9646 

1.0850  .0354 

.7353  .8665 

20 

.8261 

.7476 

50 

.6799  .8324 

.9271  .9671 

1.0786  .0329 

.7333  .8653 

10 

.8232 

.7505 

43°  00' 

.6820  .8338 

.9325  .9697 

1.0724  .0303 

.7314  .8641 

47°  00' 

.8203 

.7534 

10 

.6841  .8351 

.9380  .9722 

1.0661  .0278 

.7294  .8629 

50 

.8174 

.7563 

20 

.6862  .8365 

.9435  .9747 

1.0599  .0253 

.7274  .8618 

40 

.8145 

.7592 

30 

.6884  .8378 

.9490  .9772 

1.0538  .0228 

.7254  .8606 

30 

.8116 

.7621 

40 

.6905  .8391 

.9545  .9798 

1.0477  .0202 

.7234  .8594 

20 

.8087 

.7650 

50 

.6926  .8405 

.9601  .9823 

1.0416  .0177 

.7214  .8582 

10 

.8058 

.7679 

44°  00' 

.6947  .8418 

.9657  .9848 

1.0.355  .0152 

.7193  .8569 

46°  00' 

.8029 

.7709 

10 

.6%7  .8131 

.9713  .9874 

1.0295  .0126 

.7173  .8557 

50 

.7999 

.7738 

20 

.6988  .8444 

.9770  .9899 

1.02a5  .0101 

.7153  .8545 

40 

.7970 

.7767 

30 

.7009  .84.57 

.9827  .9924 

1.0176  .0076 

.7133  .8532 

30 

.7941 

.7796 

40 

.7030  .8469 

.9884  .9949 

1.0117  .0051 

.7112  .8520 

20 

.7912 

.7825 

50 

.7050  .8482 

.9942  .9975 

1.0058  .0025 

.7092  .8507 

10 

.7883 

.7854 

46°  00' 

.7071  .8495 

1.0000  .0000 

1.0000  .0000 

.7071  .8495 

45°  00' 

.7854 

Value  Logio 

Value   Logjo 

Value   Logjo 

Value  Logio 

Degrees 

Radians 

Cosine 

Cotangent 

Tangent 

Sine 

SLIDE-RULE 


I 


II 


(J)  (S)  (S) 


-■I  I 


i 


iiii 


Directions 

A  reasonably  accurate  slide-rule 
may  be  made  by  the  student,  for 
temporary  practice,  as  follows. 
Take  three  strips  of  heavy  stiff 
cardboard  l'^3  wide  by  &'  long; 
these  are  shown  in  cross-section  in 
(1),  (2),  (8)  above.  On  (3) 
paste  or  glue  the  adjoining  cut 
of  the  slide  rule.  Then  cut  strips 
(2)  and  (3)  accurately  along  the 
lines  marked.  Paste  or  glue  the 
pieces  together  as  shown  in  (4) 
and  (5).  Then  (5)  forms  the 
slide  of  the  slide-rule,  and  it  will 
fit  in  the  groove  in  (4)  if  the  work 
has  been  carefully  done.  Trim 
off  the  ends  as  shown  in  the  large 
cut. 


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